diff --git "a/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt" "b/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt" @@ -0,0 +1,1179 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf,len=1178 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='11789v1 [math-ph] 27 Jan 2023 A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity Lutz Angermann∗ January 30, 2023 The present work describes some extensions of an approach, originally devel- oped by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Yatsyk and the author, for the theoretical and numerical analysis of scattering and radiation effects on infinite plates with cubically polarized lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The new aspects lie on the transition to more generally shaped, two- or three-dimensional objects, which no longer necessarily have to be represented in terms a Cartesian product of real intervals, to more general nonlinearities (in- cluding saturation) and the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The paper advocates an approach that consists in transform- ing the original full-space problem for a nonlinear Helmholtz equation (as the simplest model) into an equivalent boundary-value problem on a bounded do- main by means of a nonlocal Dirichlet-to-Neumann (DtN) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It is shown that the transformed problem is equivalent to the original one and can be solved uniquely under suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Morever, the impact of the truncation of the DtN operator on the resulting solution is investigated, so that the way to the numerical solution by appropriate finite element methods is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Keywords: Scattering, radiation, nonlinear Helmholtz equation, nonlinearly polarizable medium, DtN operator, truncation AMS Subject Classification (2022): 35 J 05 35 Q 60 78 A 45 1 Introduction The present work deals with the mathematical modeling of the response of a penetrable two- or three-dimensional object (obstacle), represented by a bounded domain, to the excitation ∗Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' of Mathematics, Clausthal University of Technology, Erzstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, D-38678 Clausthal-Zellerfeld, Ger- many, lutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='angermann@tu-clausthal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='de 1 Resonant compactly supported nonlinearities January 30, 2023 by an external electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A special aspect of the paper is that, in contrast to many other, thematically comparable works, nonlinear constitutive laws of this object are in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A standard example are the so-called Kerr nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It is physically known, but also only little investigated mathematically that sufficiently strong incident fields, under certain conditions, cause effects such as frequency multiplication, which cannot occur in the linear models frequently considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' On the other hand, such effects are interest- ing in applications, which is why a targeted exploitation, for example from a numerical or optimization point of view, first requires thorough theoretical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A relatively simple mathematical model for this is a nonlinear Helmholtz equation, which results from the transition from the time-space formulation of Maxwell’s equations to the frequency-space formulation together with further simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although some interesting nonlinear effects cannot be modeled by means of a single scalar equation alone, its inves- tigation is of own importance, for example from the aspect of variable coefficients, and on the other hand its understanding is also the basis for further development, for example for systems of nonlinear Helmholtz equations, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [AY19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The latter is also the reason why we consider a splitted nonlinearity and not concentrate the nonlinearity in one term as is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Helmholtz equation with nonlinearities has only recently become the focus of mathe- matical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, problems are mainly dealt with in which the nonlinearities are globally smooth, while here a formulation as a transmission problem is used that allows less smooth transitions at the object boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, we allow more general nonlin- earities than the Kerr nonlinearities mentioned, in particular saturation effects can be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Starting from a physically oriented problem description as a full-space problem, we derive a weak formulation on a bounded domains using the well-known technique of DtN operators, and show its equivalence to the weakly formulated original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the influence of the external field only occurs indirectly in the weak formulation, we also give a second variant of the weak formulation that better clarifies this influence and which we call the input-output formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the DtN operators are non-local, their practical application (numerics) causes prob- lems, which is why a well-known truncation technique is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This raises the problem of proving the well-posedness of the reduced problem and establishing a connection (error estimate) of the solution of the reduced problem to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although these questions in the linear case have been discussed in the literature for a relatively long time, they even for the linear case seemed to have been treated only selectively and sometimes only very vaguely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The latter concerns in particular the question of the independence of the stability constant from the truncation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In this work, both stability and er- ror estimates are given for the two- and three-dimensional case, whereby a formula-based relationship between the discrete and the continuous stability constant is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Another difference to many existing, especially older works is that the present paper works with variational (weak) formulations but not with integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Unfortunately, the complete tracking of the dependence of the occurring parameters on the wave number (so- called wavenumber-independent bounds) has not yet been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It has already been mentioned that, for the linear situation, in connection with scattering problems or with problems that are formulated from the very beginning in bounded domains 2 Resonant compactly supported nonlinearities January 30, 2023 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', with impedance boundary conditions), there is an extensive and multi-threaded body of literature that is beyond the scope of this article to list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Transmission problems of the type considered here are rarely found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Nevertheless, without claiming completeness, a few works should be mentioned here that had an influence on the present results and whose bibliographies may be of help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A frequently cited work that deals with linear scattering problems in two dimensions and also served as the motivation for the present work is [HNPX11], which, however, does not discuss the dependence of the stability constant on the truncation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A number of later works by other authors quote this work, but sometimes assume results that cannot be found in the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The work that comes closest to our intentions is [Koy07], where the exterior Dirichlet boundary-value problem for the linear Helmholtz equation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In this paper, no separate, parameter-uniform stability estimate of the truncated problem is given, but the truncation error is included in the error estimate of a finite element approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A similar work is [Koy09], but in which another boundary condition at the boundary of the auxiliary domain is considered, the so-called modified DtN condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Among the more recent papers, works by Mandel [Man19], Chen, Ev´equoz & Weth [CEW21], and Maier & Verf¨urth [MV22] should be mentioned, especially because of the cited sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In his cumulative habilitation thesis, which contains further references, Mandel examines ex- istence and uniqueness questions for solutions of systems of nonlinear Helmholtz equations in the full-space case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Scattering or transmission problems are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' consider the scattering problem with quite high regularity assumptions to the superlinear nonlinearities, but without truncation approaches and not in the context of variational solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Maier & Verf¨urth, who focus mainly on multiscale aspects for a nonlinear Helmholtz equation over a bounded domain with impedance boundary conditions, give an instructive review of the literature on nonlinear Helmholtz equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The structure of the present work is based on the program outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' After the problem formulation in Section 2, the exterior auxiliary problem required for truncation is discussed, after which the weak formulation and equivalence statement follow in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Section 5 is dedicated to the existence and uniqueness of the weak solution, where in particular the assumptions on the nonlinear terms are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The final section then deals with the properties of the truncated problem – uniform (with respect to the truncation parameter) well-posedness and estimate of the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 Problem formulation Let Ω ⊂ Rd be a bounded domain with a Lipschitz boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It represents a medium with a nonlinear behaviour with respect to electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since Ω is bounded, we can choose an open Euclidean d-ball BR ⊂ Rd of radius R > supx∈Ω |x| with center in the origin such that Ω ⊂ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The complements of Ω and BR are denoted by Ωc := Rd \\ Ω Bc R := Rd \\ BR, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', the open complement of BR is denoted by B+ R := Rd \\ BR (the overbar over sets denotes their closure in Rd), and the boundary of BR, the sphere, by SR := ∂BR (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The open complement of Ω is denoted by Ω+ := Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By ν we denote the outward-pointing (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' either Ω or BR) unit normal vector on ∂Ω or SR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Trace operators will be denoted by one and the same symbol γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the concrete meaning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', 3 Resonant compactly supported nonlinearities January 30, 2023 Ω SR uinc Figure 1: The nonlinear medium Ω is excited by an incident field uinc (d = 2) traces on the common interface of an interior and exterior domain) will be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The classical direct problem of radiation and propagation of an electromagnetic field – ac- tually just one component of it – by/in the penetrable obstacle Ω is governed by a nonlinear Helmholtz equation with a variable complex-valued wave coefficient: − ∆u(x) − κ2c(x, u) u = f(x, u) for (almost) all x ∈ Rd, (1) where the wavenumber κ > 0 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The physical properties of the obstacle Ω are described by the coefficient c : Rd × C → C (physically the square of the refractive index) and the right-hand side f : Rd × C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In general, both functions are nonlinear and have the following properties: supp(1 − c(·, w)) = Ω and supp f(·, w) ⊂ Ω for all w ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2) The function 1 − c is often called the contrast function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Basically we assume that c and f are Carath´eodory functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the mapping x �→ c(x, v) is (Lebesgue-)measurable for all v ∈ C, and the mapping v �→ c(x, v) is continuous for almost all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' These two conditions imply that x �→ c(x, v(x)) is measurable for any measurable v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The same applies to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The unknown total field u : Rd → C should have the following structure: u = � urad + uinc in Ωc, utrans in Ω, (3) where urad : Ωc → C is the unknown radiated/scattered field, utrans : Ω → C denotes the unknown transmitted field, and the incident field uinc ∈ H1 loc(Ω+) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The incident field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz equation (even in the whole space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Typically it is generated either by concentrated sources located in a bounded region of Ω+ or by sources at infinity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' travalling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 1 (d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The incident plane wave, whose transmission and scattering is inves- tigated, is given by uinc(x) := αinc exp(i(Φx1 − Γx2)), x = (x1, x2)⊤ ∈ B+ R 4 Resonant compactly supported nonlinearities January 30, 2023 with amplitude αinc and angle of incidence ϕinc, |ϕinc| < π, where Φ := κ sin ϕinc is the longitudinal wave number and Γ := √ κ2 − Φ2 = κ cos ϕinc the transverse wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In polar coordinates is then uinc(r, ϕ) = αinc exp(i(Φr cos ϕ − Γr sin ϕ)) = αinc exp(iκr(sin ϕinc cos ϕ − cos ϕinc sin ϕ)) = αinc exp(iκr sin(ϕinc − ϕ)), (r, ϕ) ∈ B+ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The radiated/scattered field urad should satisfy an additional condition, the so-called Som- merfeld radiation condition: lim |x|→∞ |x|(d−1)/2 � ˆx · ∇urad − iκurad� = 0 (4) uniformly for all directions ˆx := x/|x|, where ˆx·∇urad denotes the derivative of urad in radial direction ˆx, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [CK13, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='7) for d = 3, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='96) for d = 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Physically, the condition (4) allows only outgoing waves at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' mathematically it guaranties the uniqueness of the solution uscat : B+ R → C of the following exterior Dirichlet problem −∆uscat − κ2uscat = 0 in B+ R, uscat = fSR on SR, lim |x|→∞ |x|(d−1)/2 � ˆx · ∇uscat − iκuscat� = 0, (5) where fSR : SR → C is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We mention that, in the context of classical solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' uscat ∈ C2(B+ R)) to problem (5), Rellich [Rel43] has shown that the condition (4) can be weakened to the following integral version: lim |x|→∞ � SR ��ˆx · ∇uscat − iκuscat��2 ds(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the context of weak solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' uscat ∈ H1 loc(B+ R)), an analogous equivalence statement can be found in [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3 The exterior problem in Bc R For a given fSR ∈ C(SR) and d = 3, the unique solvability of problem (5) in C2(B+ R)∩C(Bc R) is proved, for example, in [CK13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, if fSR is smoother, say fSR ∈ C∞(SR), then the normal derivative of uscat on the boundary SR is a well-defined continuous function [CK13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' These assertions remain valid in the case d = 2, see [CK13, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Therefore, by solving (5) for given fSR ∈ C∞(SR), a mapping can be introduced that takes the Dirichlet data on SR to the corresponding Neumann data on SR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' fSR �→ TκfSR := ˆx · ∇uscat�� SR , (6) see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [CK19, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 5 Resonant compactly supported nonlinearities January 30, 2023 Furthermore, it is well-known that the mapping Tκ can be extended to a bounded linear operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) for any |s| ≤ 1/2 [CWGLS12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='31] (we keep the notation already introduced for this continued operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This operator is called the Dirichlet-to-Neumann operator, in short DtN operator, or capacity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the problem (5) is considered in a spherical exterior domain, an explicit series represen- tation of the solution is available using standard separation techniques in polar or spherical coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The term-by-term differentiation of this series thus also provides a series representation of the image of Tκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The solution of the problem (5) in the two-dimensionsional case (here with uscat replaced by u) is given by [Mas87, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], [KG89, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (30)]: u(x) = u(rˆx) = u(r, ϕ) = � n∈Z H(1) n (κr) H(1) n (κR) fn(R)Yn(ˆx) = � n∈Z H(1) n (κr) H(1) n (κR) fn(R)Yn(ϕ), x = rˆx ∈ Sr, r > R, ϕ ∈ [0, 2π] (7) (identifying u(x) with u(r, ϕ) and Yn(ˆx) with Yn(ϕ) for x = rˆx = r(cos ϕ, sin ϕ)⊤), where (r, ϕ) are the polar coordinates, H(1) n are the cylindrical Hankel functions of the first kind of order n [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2]1, Yn are the circular harmonics defined by Yn(ϕ) = einϕ √ 2π , n ∈ Z, fn(R) are the Fourier coefficients of fSR defined by fn(R) := (fSR(R·), Yn)S1 = � S1 fSR(Rˆx)Yn(ˆx)ds(ˆx) = � 2π 0 fSR(R, ϕ)Yn(ϕ)dϕ, (8) and ds(ˆx) is the Lebesgue arc length element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we formally differentiate the representation (7) with respect to r to obtain the outward normal derivative of u: ˆx · ∇u(x) = ∂u ∂r (rˆx) = κ � n∈Z H(1)′ n (κr) H(1) n (κR) fn(R)Yn(ˆx), x = rˆx ∈ Sr, r > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Setting fR := u|SR and letting x in this representation approach the boundary SR, we can formally define the (extended) DtN operator by Tκu(x) := 1 R � n∈Z Zn(κR)un(R)Yn(ˆx), x = Rˆx ∈ SR, (9) where Zn(ξ) := ξ H(1)′ n (ξ) H(1) n (ξ) , 1Instead of (4) [Mas87] considered the ingoing Sommerfeld condition and thus obtained a representation in terms of the cylindrical Hankel functions of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that H(2) n (−ξ) = −(−1)nH(1) n (ξ) [DLMF22, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6 Resonant compactly supported nonlinearities January 30, 2023 and un(R) are the Fourier coefficients of u|SR analogously to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The admissibility of this procedure has been proven in many sources in the classical context, for example [CK19, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the present case, in the paper [Ern96, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1] it was shown that the operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Ernst’s result was extended to all s ≥ 0 in [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the case d = 3, the solution of the problem (5) is given by [KG89, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (33)]: u(x) = u(rˆx) = u(r, ϕ, θ) = � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ˆx) = � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ϕ, θ), x ∈ Sr, r > R, (ϕ, θ) ∈ [0, 2π] × [0, π] (10) (identifying u(x) with u(r, ϕ, θ) and Y m n (ˆx) with Y m n (ϕ, θ) for x = rˆx = r(cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where (r, ϕ, θ) are the spherical coordinates, h(1) n are the spherical Hankel functions of the first kind of order n [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='47], Y m n are the spherical harmonics defined by Y m n (ϕ, θ) = � 2n + 1 4π (n − |m|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (n + |m|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' P |m| n (cos θ)eimϕ, n ∈ N0, |m| ≤ n, (identifying Y m n (ˆx) with Y m n (ϕ, θ) for ˆx = (cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where P m n are the associated Legendre functions of the first kind [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='21], f m n (R) are the Fourier coefficients defined by f m n (R) = (fSR(R·), Y m n )S1 = � S1 fSR(Rˆx)Y m n (ˆx)ds(ˆx) = � 2π 0 � π 0 fSR(R, ϕ, θ)Y m n (ϕ, θ) sin θdθdϕ, (11) and ds(ˆx) is the Lebesgue surface area element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proceeding as in the two-dimensional case, we get ˆx · ∇u(x) = ∂u ∂r (rˆx) = κ � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ˆx), x = rˆx ∈ Sr, r > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Setting fR := u|SR and letting r → R, we can define the (extended) DtN operator by Tκu(x) = 1 R � n∈N0 � |m|≤n zn(κR)um n (R)Y m n (ˆx), x = Rˆx ∈ SR, (12) where zn(ξ) := ξ h(1)′ n (ξ) h(1) n (ξ) , 7 Resonant compactly supported nonlinearities January 30, 2023 and um n (R) are the Fourier coefficients of u|SR analogously to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The admissibility of this procedure is proved in [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='15] or [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the present situation there is a boundedness result for d = 3 analogous to [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] in [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In summary, the following statement applies to both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The DtN operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A more refined analysis of the DtN operator in the case s = 0 results in a sharp estimate of the its norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the wavenumber [BSW16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4]: Given κ0 > 0, there exists a constant C > 0 independent of κ such that ∥Tκv∥−1/2,2,SR ≤ Cκ∥v∥1/2,2,SR for all v ∈ H1 loc(B+ R) and κ ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The result from [BSW16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4] applies to more general domains, for the present situation it already follows from the proof of Lemma 23 (see the estimates (46), (47) for s = 0, where the bounds do not depend on N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' At the end of this section we give a collection of some properties of the coefficient functions in the representations (9), (12) which will be used in some of the subsequent proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For all ξ > 0, the following holds: −n ≤ Re Zn(ξ) ≤ −1 2, 0 < Im Zn(ξ) < ξ for all |n| ∈ N, −1 2 ≤ Re Z0(ξ) < 0, ξ < Im Z0(ξ), −(n + 1) ≤ Re zn(ξ) ≤ −1, 0 < Im zn(ξ) ≤ ξ for all n ∈ N, Re z0(ξ) = −1, Im z0(ξ) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the case d = 2, the estimates can be found in [SW07, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The other estimates can be found in [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], see also [SW07, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='22), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='23)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although only 0 ≤ Im zn(ξ) is specified in the formulation of the cited theorem, the strict positivity follows from the positivity of the function qℓ in [N´ed01, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34)], as has been mentioned in [MS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For all ξ > 0, the following holds: |Zn(ξ)|2 ≤ (1 + n2)(1 + |ξ|2) for all |n| ∈ N, |zn(ξ)|2 ≤ (1 + n2)(2 + |ξ|2) for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimates of the real and imaginary parts of Zn from Lemma 4 immediately imlpy that 1 1 + n2|Zn(ξ)|2 = 1 1 + n2 � | Re Zn(ξ)|2 + | Im Zn(ξ)|2� ≤ 1 1 + n2 � n2 + |ξ|2� ≤ 1 + |ξ|2 1 + n2 ≤ 1 + |ξ|2, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since H(1) −n(ξ) = (−1)nH(1) n (ξ), n ∈ N [DLMF22, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2)], the estimate is also valid for n such that −n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 8 Resonant compactly supported nonlinearities January 30, 2023 Analogously we obtain from Lemma 4 that 1 1 + n2|zn(ξ)|2 = 1 1 + n2 � | Re zn(ξ)|2 + | Im zn(ξ)|2� ≤ 1 1 + n2 � (1 + n)2 + |ξ|2� ≤ 2 + |ξ|2 1 + n2 ≤ 2 + |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4 Weak formulations of the interior problem Now we turn to the consideration of the problem (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the classical setting it can be formulated as follows: Given uinc ∈ H1 loc(Ω+), determine the transmitted field utrans : Ω → C and the radiated/scattered field urad : Ωc → C satisfying −∆utrans − κ2c(·, utrans) utrans = f(·, utrans) in Ω, −∆urad − κ2urad = 0 in Ω+, utrans = urad + uinc on ∂Ω, ν · ∇utrans = ν · ∇urad + ν · ∇uinc on ∂Ω (13) and the radiation condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that the incident field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz equation in Ω+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the second equation in (13) can be replaced by − ∆u − κ2u = f inc in Ω+, (14) where f inc : Ω+ → C is an eventual source density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For simplicity we do not include the case of a nontrivial source density in our investigation, but the subsequent theory can be easily extended by adding an appropriate linear functional, say ℓsrc, on the right-hand side of the obtained weak formulations (see (15) or (19) later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In order to give a weak formulation of (13) with the modification (14) in the case f inc = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we introduce the (complex) linear function spaces H1 comp(Ω+) := � v ∈ H1(Ω+) : supp v is compact � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' VRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v|Ω+ ∈ H1 loc(Ω+) : γv|Ω = γv|Ω+ on ∂Ω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' WRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v|Ω+ ∈ H1 comp(Ω+) : γv|Ω = γv|Ω+ on ∂Ω} (note the comment at the beginning of Section 2 on the notation for trace operators) and multiply the first equation of (13) by the restriction v|Ω of an arbitrary element v ∈ VRd and (14) by the restriction v|Ω+ of v ∈ VRd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' and integrate py parts: (∇utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)Ω − (ν · ∇utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)∂Ω − κ2(c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' utrans)utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω = (f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' utrans),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (∇u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)Ω − (ν · ∇u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)∂Ω+ − κ2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9 Resonant compactly supported nonlinearities January 30, 2023 Here we use the notation, for any domain M ⊂ Rd with boundary ∂M and appropriately defined functions on M or ∂M, (∇w, ∇v)M := � M ∇w · ∇vdx, (w, v)M := � M wvdx, (w, v)∂M := � ∂M wvds(x) (the overbar over functions denotes complex conjugation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Taking into consideration the last transmission condition in (13), the relationsship ν|Ω = −ν|Ω+, and the fact that the last but one transmission condition in (13) is included in the definition of the space VRd, we define a bivariate nonlinear form on VRd × WRd by aRd(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)Ω+ − κ2(c(·, w)w, v)Rd, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [Wlo87, Example 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Given uinc ∈ H1 loc(Ω+), a weak solution to the problem (1)–(4) is defined as an element u ∈ VRd that has the structure (3), satisfies the variational equation aRd(u, v) = (f(·, u), v)Rd for all v ∈ WRd (15) and the Sommerfeld radiation condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A second weak formulation can be obtained if we do not replace the second Helmholtz equation in (13) by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the first step in the derivation of the weak formulation reads as (∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω, (∇urad, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω+ − κ2(urad, v)Ω+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The last transmission condition in (13) allows to rewrite the first equation as (∇utrans, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω + (ν · ∇uinc, ∇v)∂Ω, leading to the weak formulation (∇u0, ∇v)Ω+(∇u0, ∇v)Ω+−κ2(c(·, u0)u0, v)Rd = (f(·, u0), v)Rd+(ν·∇uinc, v)∂Ω for all v ∈ WRd (16) with respect to the structure u0 := � urad in Ωc, utrans in Ω, (17) where urad ∈ H1 loc(Ω+), utrans ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The advantage of this formulation is that it clearly separates the unknown and the known parts of the fields, so we call this formulation the input-output formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The disadvantage 10 Resonant compactly supported nonlinearities January 30, 2023 is that the natural function space of the solution u0 is not a linear space due to the last but one transmission condition in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Instead of the problem (1)–(4) we want to solve an equivalent problem in the bounded domain BR, that is, we define V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\\Ω ∈ H1(BR \\ Ω) : γv|Ω = γv|BR\\Ω on ∂Ω} and look for an element u ∈ V such that −∆utrans − κ2c(·, utrans) u = f(·, utrans) in Ω, −∆u − κ2u = 0 in BR \\ Ω, utrans = urad + uinc on ∂Ω, ν · ∇utrans = ν · ∇urad + ν · ∇uinc on ∂Ω, ˆx · ∇urad = Tκurad on SR (18) formally holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now the weak formulation of problem (18) reads as follows: Find u ∈ V such that (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR = (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR (19) for all v ∈ V holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The weak formulations (15) and (19) of the problems (1)–(4) and (18), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First let u ∈ V (Rd) be a weak solution to (1)–(4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it satisfies (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then its restriction to BR belongs to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To demonstrate that this restriction satisfies the weak formulation (19), we construct the radiating solution uBc R′ of the homogeneous Helmholtz equation outside of a smaller ball BR′ such that Ω ⊂ BR′ ⊂ BR and uBc R′ ��� SR′ = (u − uinc)|SR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This solution can be constructed in the form of a series expansion in terms of Hankel functions as explained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By elliptic regularity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='16], [Eva15, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1]), the solution of this problem satisfies the Helmholtz equation in Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, by uniqueness [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5], it coincides with u − uinc = urad in Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we choose a finite partition of unity covering BR, denoted by {ϕj}J [Wlo87, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2], such that its index set J can be decomposed into two disjoint subsets J1, J2 as follows: BR′ ⊂ int � � j∈J1 supp ϕj � , � j∈J1 supp ϕj ⊂ BR, � j∈J2 supp ϕj ⊂ Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For example, we can choose {ϕj}J1 to consist of one element, say ϕ1, namely the usual mollifier function with support B′, where the open ball B′ (centered at the origin) lies between BR′ and BR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' BR′ ⊂ B′ = int (supp ϕ1), supp ϕ1 ⊂ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the second part consists of a finite open covering of the spherical shell BR \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 11 Resonant compactly supported nonlinearities January 30, 2023 Then we take, for any v ∈ V , the product v1 := v � j∈J1 ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is an element of V , too, with support in BR, and it can be continued by zero to an element of W(Rd) (keeping the notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence we can take it as a test function in the weak formulation (15) and obtain aRd(u, v1) = (f(·, u), v1)Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is equal to (∇u, ∇v1)Ω + (∇u, ∇v1)BR\\Ω − κ2(c(·, u)u, v1)BR − (Tκu, v1)SR = (f(·, u), v1)BR − (Tκuinc, v1)SR + (ˆx · ∇uinc, v1)SR due to the properties of the support of v1 (in particular, all terms “living” on SR are equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the homogeneous Helmholtz equation is satisfied in � j∈J2 supp ϕj ⊂ Bc R′, we can proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We continue the test function v2 := v � j∈J2 ϕj by zero into the complete ball BR and have (f(·, u), v2)BR\\Ω = 0 = (−∆u − κ2u, v2)BR\\BR′ = (∇u, ∇v2)BR\\BR′ − κ2(u, v2)BR\\BR′ − (ν · ∇u, v2)∂(BR\\BR′) = (∇u, ∇v2)BR\\BR′ − κ2(u, v2)BR\\BR′ − (ˆx · ∇u, v2)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now, taking into consideration the properties of the support of v2, we easily obtain the following relations: (∇u, ∇v2)BR\\BR′ = (∇u, ∇v2)Ω + (∇u, ∇v2)BR\\Ω, (u, v2)BR\\BR′ = (c(·, u)u, v2)BR, (ˆx · ∇u, v2)SR = (ˆx · ∇urad, v2)SR + (ˆx · ∇uinc, v2)SR = (Tκurad, v2)SR + (ˆx · ∇uinc, v2)SR = (Tκu, v2)SR − (Tκuinc, v2)SR + (ˆx · ∇uinc, v2)SR, where the treatment of the last term makes use of the construction of the Dirichlet-to- Neumann map Tκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Adding both relations and observing that v = v1+v2, we arrive at the variational formulation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Conversely, let u ∈ V be a solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To continue it into Bc R, similar to the first part of the proof we construct the radiating solution uBc R of the Helmholtz equation outside BR such that uBc R �� SR = (u − uinc)|SR and set u := uBc R + uinc in B+ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence we have that Tκu = ∂uBc R ∂ˆx + Tκuinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we take an element v ∈ W(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Its restriction to BR is an element of V and thus can be taken as a test function in (19): (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR = (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (20) 12 Resonant compactly supported nonlinearities January 30, 2023 Since v has a compact support, we can choose a ball B ⊂ Rd centered at the origin such that BR ∪ supp v ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The homogeneous Helmholtz equation is obviously satisfied in the spherical shell B \\ BR: −∆uBc R − κ2uBc R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We multiply this equation by the complex conjugate of the test function v ∈ V , then integrate over the shell, and apply the first Green’s formula: (∇uBc R, ∇v)B\\BR − κ2(uBc R, v)B\\BR − (ν · ∇uBc R, v)∂(B\\BR) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we observe that (∇uBc R, ∇v)B\\BR = (∇uBc R, ∇v)B+ R, (uBc R, v)B\\BR = (uBc R, v)B+ R, (ν · ∇uBc R, v)∂(B\\BR) = −(ˆx · ∇uBc R, v)SR = −(Tκu − Tκuinc, v)SR where the minus sign in the last line results from the change in the orientation of the outer normal (once w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the shell, once w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' BR) and the construction of uBc R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So we arrive at (∇uBc R, ∇v)B+ R − κ2(uBc R, v)B+ R + (Tκu, v)SR = (Tκuinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (21) Finally, since the incident field satisfies the homogeneous Helmholtz equation in the spherical shell, too, we see by an analogous argument that the variational equation (∇uinc, ∇v)B+ R − κ2(uinc, v)B+ R = −(ˆx · ∇uinc, v)SR (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Adding the variational equations (20) – (22), we arrive at the variational formulation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 5 Existence and uniqueness of a weak solution In this section we investigate the existence and uniqueness of the weak solution of the interior problem (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We define the sesquilinear form a(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)BR\\Ω − κ2(w, v)BR − (Tκw, v)SR for all w, v ∈ V, (23) the nonlinear form n(w, v) := κ2(c(·, w) − 1)w, v)BR + (f(·, w), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR (24) and reformulate (19) as follows: Find u ∈ V such that a(u, v) = n(u, v) for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (25) On the space V , we use the standard seminorm and norm: |v|V := � ∥∇v∥2 0,2,Ω + ∥∇v∥2 0,2,BR\\Ω �1/2 , ∥v∥V := � |v|2 V + ∥v∥2 0,2,BR �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (26) 13 Resonant compactly supported nonlinearities January 30, 2023 For κ > 0, the following so-called wavenumber dependent norm on V is also common: ∥v∥V,κ := � |v|2 V + κ2∥v∥2 0,2,BR �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (27) It is not difficult to verify that the standard norm and the wavenumber dependent norm are equivalent on V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it holds C−∥v∥V ≤ ∥v∥V,κ ≤ C+∥v∥V for all v ∈ V, (28) where the equivalence constants depend on κ in the following way: C− := min{1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ} and C+ := max{1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We now proceed to examine the linear aspects of the problem (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The sesquilinear form a is bounded on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Applying to each addend in the definition of a the appropriate Cauchy-Bunyakovsky- Schwarz inequality, we obtain |a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + ∥Tκw∥−1/2,2,SR∥v∥1/2,2,SR for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' According to Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 the DtN operator Tκ is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant CTκ > 0 such that ∥Tκw∥−1/2,2,SR ≤ CTκ∥w∥1/2,2,SR for all w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It remains to apply a trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='37]: |a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 tr∥w∥1,2,BR\\Ω∥v∥1,2,BR\\Ω ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 tr∥w∥V ∥v∥V ≤ min{(max{1, κ2} + CTκC2 tr)∥w∥V ∥v∥V , (1 + CTκC2 tr)∥w∥V,κ∥v∥V,κ} for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Given κ0 > 0 and R0 > 0, assume that κ ≥ κ0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3) and R ≥ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, κ0 ≥ 1 is required for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the sesquilinear form a satisfies a G˚arding’s inequality of the form Re a(v, v) ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From the definitions of a and the wavenumber dependent norm it follows immediately that Re a(v, v) = ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR − Re (Tκv, v)SR ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR + CR−1∥v∥2 0,2,SR ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR, where the first estimate follows from [MS10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3] with a constant C > 0 depending soleley on κ0 > 0 and R0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 14 Resonant compactly supported nonlinearities January 30, 2023 Next we discuss the solvability and stability of the problem (25) for the case that the right- hand side is just an antilinear continuous functional ℓ : V → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The linear problem of finding u ∈ V such that a(u, v) = ℓ(v) for all v ∈ V (29) holds can be formulated equivalently as an operator equation in the dual space V ∗ of V consisting of all continuous antilinear functionals from V to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Namely, if we define the linear operator A : V → V ∗ by Aw(v) := a(w, v) for all w, v ∈ V, (30) problem (29) is equivalent to solving the operator equation Au = ℓ (31) for u ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that A is a bounded operator by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, the problem (31) is uniquely solvable for any ℓ ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The basic ideas of the proof are taken from the proof of [MS10, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the embedding of V into L2(BR) is compact by the compactness theorem of Rellich–Kondrachov [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27] together with Tikhonov’s product theorem [KN63, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], the com- pact perturbation theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34] together with Lemma 9 imply that the Fredholm alternative [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27] holds for the equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence it is sufficient to demonstrate that the homogeneous adjoint problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 43]) of finding u ∈ V such that a(v, u) = 0 holds for all v ∈ V only allows for the trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So suppose u ∈ V is a solution of the homogeneous adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We take v := u and consider the imaginary part of the resulting equation: 0 = Im a(u, u) = − Im (Tκu, u)SR = Im (Tκu, u)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then [MS10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3] implies u = 0 on SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then u satisfies the variational equation (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(u, v)BR = 0 for all v ∈ V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it is a weak solution of the homogeneous interior transmission Neumann problem for the wave equation on BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' On the other hand, u can be extended to the whole space Rd by zero to an element ˜u ∈ V (Rd), and this element can be interpreted as a weak solution of a homogeneous full-space transmission problem, for instance in the sense of [TW93, Problem (P)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then it follows from [TW93, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] that ˜u = 0 und thus u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since a Fredholm operator has a closed image [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 33], it follows from the Open Mapping Theorem and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2]) that the inverse operator A−1 is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant C(R, κ) > 0 such that ∥u∥V,κ = ∥A−1ℓ∥V,κ ≤ C(R, κ)∥ℓ∥V ∗ for all ℓ ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 15 Resonant compactly supported nonlinearities January 30, 2023 Then it holds 1 C(R, κ) ≤ ∥ℓ∥V �� ∥u∥V,κ = sup v∈V \\{0} |ℓ(v)| ∥u∥V,κ∥v∥V,κ = sup v∈V \\{0} |a(u, v)| ∥u∥V,κ∥v∥V,κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This estimate proves the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, the sesquilinear form a satisfies an inf-sup condition: β(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |a(w, v)| ∥w∥V,κ∥v∥V,κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we turn to the nonlinear situation and concretize the assumptions regarding the Cara- th´eodory functions c and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Let pf ∈ � [2, ∞), d = 2, [2, 6], d = 3, and assume there exist nonnegative functions mf, gf ∈ L∞(Ω) such that |f(x, ξ)| ≤ mf(x)|ξ|pf−1 + gf(x) for all (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then vf(·, w) ∈ L1(Ω) for all w, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since f is a Carath´eodory function, the composition f(·, w) is measurable and it sufficies to estimate the integral of |vf(·, w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, it suffices to consider the term mfv|w|pf−1 in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By H¨older’s inequality for three functions, it holds that ∥vf(·, w)∥0,1,Ω ≤ ∥mf∥0,∞,Ω∥v∥0,pf,Ω∥wpf−1∥0,q,Ω with 1 pf + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Lpf-norm of v is bounded thanks to the embedding V |Ω ⊂ Lpf(Ω) for the allowed values of pf [AF03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since |wpf−1|q = |w|p f, the Lq-norm of wpf−1 is bounded by the same reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Let pc ∈ � [2, ∞), d = 2, [2, 6], d = 3, and assume there exist nonnegative functions mc, gc ∈ L∞(Ω) such that |c(x, ξ) − 1| ≤ mc(x)|ξ|pc−2 + gc(x) for all (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then zv(c(·, w) − 1) ∈ L1(Ω) for all z, w, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Similar to the proof of Lemma 12 it is sufficient to consider the term mczv|w|pc−2 in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By H¨older’s inequality for four functions, it holds that ∥zv(c(·, w) − 1)∥0,1,Ω ≤ ∥mc∥0,∞,Ω∥z∥0,pc,Ω∥v∥0,pc,Ω∥wpc−2∥0,q,Ω with 2 pc + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Lpc-norms of z, v are bounded thanks to the embedding theorem [AF03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since |wpc−2|q = |w|p c, the Lq-norm of wpc−2 is bounded by the same reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 16 Resonant compactly supported nonlinearities January 30, 2023 Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 12 and Lemma 13, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', the following estimates hold for all z, w, v ∈ V : |(f(·, w), v)Ω| ≤ C pf emb∥mf∥0,∞,Ω∥w∥ pf−1 1,2,Ω∥v∥1,2,Ω + � |Ω|d ∥gf∥0,∞,Ω∥v∥0,2,Ω, (32) |((c(·, w) − 1)z, v)Ω| ≤ Cpc emb∥mc∥0,∞,Ω∥w∥pc−2 1,2,Ω∥z∥1,2,Ω∥v∥1,2,Ω + ∥gc∥0,∞,Ω∥z∥0,2,Ω∥v∥0,2,Ω, (33) where |Ω|d is the d-volume of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Replace v by v in Lemmata 12, 13 to get the first addend of the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimate of the second addend is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' An important example for the nonlinearities is c(x, ξ) := � 1, (x, ξ) ∈ Ω+ × C, ε(L)(x) + α(x)|ξ|2, (x, ξ) ∈ Ω × C, with given ε(L), α ∈ L∞(Ω), and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Here pc = 4, which is within the range of validity of Lemma 13, and mc = |α|, gc = |ε(L) − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimates from Corollary 14 show that the first two terms on the right-hand side of the variational equation (25) can be considered as values of nonlinear mappings from V to V ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we can define ℓcontr : V → V ∗ by ⟨ℓcontr(w), v⟩ := κ2(c(·, w) − 1)w, v)Ω, ℓsrc : V → V ∗ by ⟨ℓsrc(w), v⟩ := (f(·, w), v)Ω for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore, if uinc ∈ H1 loc(Ω+) is such that additionally ∆uinc belongs to L2,loc(Ω+) (where ∆uinc is understood in the distributional sense), the last two terms on the right-hand side of (24) form an antilinear continuous functional on ℓinc ∈ V ∗: ⟨ℓinc, v⟩ := (ˆx · ∇uinc − Tκuinc, v)SR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is a consequence of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 and the estimates before the trace theorem [KA21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence ∥ℓinc∥V ∗ ≤ ˜Ctr[∥∆uinc∥0,2,BR\\Ω + ∥uinc∥0,2,BR\\Ω] + CTκC2 tr∥uinc∥1,2,BR\\Ω, where ˜Ctr is the norm of the trace operator defined in [KA21, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='39)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, it is more intuitive to utilize the estimate ∥ℓinc∥V ∗ ≤ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (34) The reason for this is that the bound can be interpreted as a measure of the deviation of the function uinc from a radiating solution of the corresponding Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In other words: If the function uinc satisfies the boundary value problem (5) with fSR := uinc|SR, then the functional ℓinc is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 17 Resonant compactly supported nonlinearities January 30, 2023 Consequently, setting F(w) := ℓcontr(w) + ℓsrc(w) + ℓinc for all w ∈ V, we obtain a nonlinear operator F : V → V ∗, and the problem (25) is then equivalent to the operator equation Au = F(u) in V ∗, and further, by Lemma 11, equivalent to the fixed-point problem u = A−1F(u) in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (35) In order to prove the subsequent existence and uniqueness theorem, we specify some addi- tional properties of the nonlinearities c and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The functions c and f are said to generate locally Lipschitz continuous Ne- mycki operators in V if the following holds: For some parameters pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' pf ∈ � [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' d = 3,,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exist Carath´eodory functions Lc : Ω×C×C → (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞) and Lf : Ω×C×C → (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞) such that the composition operators Ω × V × V → Lqc(Ω) : (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) �→ Lc(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Ω × V × V → Lqf(Ω) : (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) �→ Lf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) are bounded for qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' qf > 0 with 3 pc + 1 qc = 2 pf + 1 qf = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' and |c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ) − c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)| ≤ Lc(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)|ξ − η|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ) − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)| ≤ Lf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)|ξ − η| (36) for all (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η) ∈ Ω × C × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If the nonlinearities c and f generate locally Lipschitz continuous Nemycki operators in the sense of the above Definition 16, the assumptions of Lemmata 12, 13 can be replaced by the requirement that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf /(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' similar to the proofs of the two lemmata mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we have that ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥v(f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w) − f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf))∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥vLf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)|w − wf|∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥w − wf∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(∥w∥V + ∥wf∥V ) � ∥v∥V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥zv(c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w) − c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc))∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥zvLc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)|w − wc|∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥z∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥z∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥w − wc∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(∥w∥V + ∥wc∥V )] ∥z∥V ∥v∥V with 1 pf + 1 ˜qf = 1 and 2 pc + 1 ˜qc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18 Resonant compactly supported nonlinearities January 30, 2023 Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, let the functions c and f generate locally Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore let uinc ∈ H1 loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If there exist numbers ̺ > 0 and LF ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' β(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ)) such that the following two conditions κ2 [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wf∥V ) � (37) + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR ≤ ̺β(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ2 [∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω̺ + ∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ LF (38) are satisfied for all w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v ∈ Kcl ̺ := {v ∈ V : ∥v∥V ≤ ̺},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' then the problem (35) has a unique solution u ∈ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we mention that Kcl ̺ is a closed nonempty subset of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Next we show that A−1F(Kcl ̺ ) ⊂ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To this end we make use of the estimates given in the proof of Remark 17 and obtain ∥F(w)∥V ∗ ≤ ∥ℓcontr(w)∥V ∗ + ∥ℓsrc(w)∥V ∗ + ∥ℓinc∥V ∗ ≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥w∥V + � ∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V ) � + ∥ℓinc∥V ∗ ≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V ) � + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence the assumption (37) implies ∥A−1F(w)∥V ≤ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It remains to show that the mapping A−1F is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the consideration of the contrast term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From the elementary decomposition (c(·, w) − 1)w − (c(·, v) − 1)v = (c(·, w) − c(·, v))w + (c(·, v) − 1)(w − v) we see that ∥ℓcontr(w) − ℓcontr(v)∥V ∗ ≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ∥w∥V + κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥V ] ∥w − v∥V ≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ̺ + κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V ≤ κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimate of the source term follows immediately from the properties of f: ∥ℓsrc(w) − ℓsrc(v)∥V ∗ ≤ ∥Lf(·, w, v)∥0,qf,Ω∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 19 Resonant compactly supported nonlinearities January 30, 2023 From ∥F(w) − F(v)∥V ∗ ≤ ∥ℓcontr(w) − ℓcontr(v)∥V ∗ + ∥ℓsrc(w) − ℓsrc(v)∥V ∗ and assumption (38) we thus obtain ∥F(w) − F(v)∥V ∗ ≤ LF∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In summary, Banach’s fixed point theorem can be applied (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [Eva15, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1]) and we conclude that the problem (35) has a unique solution u ∈ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If we introduce the function space ˜V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\\Ω ∈ H1(BR \\ Ω)} equipped with the norm ∥v∥ ˜V := � ∥v∥2 1,2,Ω + ∥v∥2 1,2,BR\\Ω �1/2 for all v ∈ ˜V , the ball Kcl ̺ appearing in the above theorem can be interpreted as a ball in ˜V of radius ̺ with center in u0 := � 0 in Ω, −uinc in BR \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed, for u of the form (3), it holds that ∥u − u0∥2 ˜V = ∥utrans∥2 1,2,Ω + ∥urad + uinc∥2 1,2,BR\\Ω = ∥u∥2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This means that the influence of the incident field uinc on the radius ̺ in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18 depends only on the deviation of uinc from a radiating field measured by ∥ℓinc∥V ∗, but not directly on the intensity of uinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In other words, if the incident field uinc is radiating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', it also satisfies the Sommerfeld radiation condition (4) and thus ℓinc = 0), the radius ̺ does not depend on uinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, uinc can be a strong field, which is important for the occurence of generation efffects of higher harmonics [AY19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 19 (Example 15 continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The identity c(·, ξ) − c(·, η) = α (|ξ|2 − |η|2) = α (|ξ| + |η|)(|ξ| − |η|) for all ξ, η ∈ C and the inequality ||ξ| − |η|| ≤ |ξ − η| show that |c(·, ξ) − c(·, η)| ≤ |α|(|ξ| + |η|)|ξ − η| holds, hence we can set Lc(·, ξ, η) := |α|(|ξ| + |η|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' With pc = qc = 4, c generates a locally Lipschitz continuous Nemycki operator in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore we may choose wc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then: ∥c(·, wc) − 1∥0,˜qc,Ω = ∥ε(L) − 1∥0,2,Ω, ∥Lc(·, w, v)∥0,qc,Ω = ∥α(|w| + |v|)∥0,4,Ω ≤ ∥αw∥0,4,Ω + ∥αv∥0,4,Ω ≤ ∥α∥0,∞,Ω [∥w∥0,4,Ω + ∥v∥0,4,Ω] ≤ Cemb∥α∥0,∞,Ω [∥w∥V + ∥v∥V ] , ∥Lc(·, w, wc)∥0,qc,Ω = ∥αw∥0,4,Ω ≤ Cemb∥α∥0,∞,Ω∥w∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 20 Resonant compactly supported nonlinearities January 30, 2023 Hence the validity of the following conditions is sufficient for (37), (38): κ2 � ∥ε(L) − 1∥0,2,Ω + Cemb∥α∥0,∞,Ω̺2� ̺ + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ), κ2 � ∥ε(L) − 1∥0,2,Ω + 3Cemb∥α∥0,∞,Ω̺2� ≤ LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A consideration of these condition shows that there can be different scenarios for which they can be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, one of the smallness requirements concerns the product ∥α∥0,∞,Ω̺3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 20 (saturated Kerr nonlinearity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Another important example for the nonlineari- ties is [Akh98] c(x, ξ) := � 1, (x, ξ) ∈ Ω+ × C, ε(L)(x) + α(x)|ξ|2/(1 + γ|ξ|2), (x, ξ) ∈ Ω × C, with given ε(L), α ∈ L∞(Ω), saturation parameter γ > 0, and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Based on the identity |ξ|2 1 + γ|ξ|2 − |η|2 1 + γ|η|2 = (1 + γ|η|2)|ξ|2 − (1 + γ|ξ|2)|η|2 (1 + γ|ξ|2)(1 + γ|η|2) = |ξ|2 − |η|2 (1 + γ|ξ|2)(1 + γ|η|2) for all ξ, η ∈ C we obtain ���� |ξ|2 1 + γ|ξ|2 − |η|2 1 + γ|η|2 ���� = (|ξ| + |η|) ||ξ| − |η|| (1 + γ|ξ|2)(1 + γ|η|2) ≤ (|ξ| + |η|)|ξ − η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence on Ω we arrive at the same Lipschitz function as in the previous Example 19, that is Lc(x, ξ, η) := � 0, (x, ξ, η) ∈ Ω+ × C × C, |α|(|ξ| + |η|), (x, ξ, η) ∈ Ω × C × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, since c(x, wc) = c(x, 0) = � 0, (x, ξ) ∈ Ω+ × C, ε(L), (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we get the same sufficient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6 The modified boundary value problem Since the exact DtN operator is represented as an infinite series (see (9), (12)), it is practically necessary to truncate this nonlocal operator and consider only finite sums Tκ,Nu(x) := 1 R � |n|≤N Zn(κR)un(R)Yn(ˆx), x = Rˆx ∈ SR ⊂ R2, (39) Tκ,Nu(x) = 1 R N � n=0 � |m|≤n zn(κR)um n (R)Y m n (ˆx), x = Rˆx ∈ SR ⊂ R3 (40) 21 Resonant compactly supported nonlinearities January 30, 2023 for some N ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The map Tκ,N is called the truncated DtN operator, and N is the truncation order of the DtN operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The replacement of the exact DtN operator Tκ in the problem (18) by the truncated DtN operator Tκ,N introduces a perturbation, hence we have to answer the question of existence and uniqueness of a solution to the following problem: Find uN ∈ V such that aN(uN, v) = nN(uN, v) for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (41) holds, where aN and nN are the forms defined by (23), (24) with Tκ replaced by Tκ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The next result is the counterpart to Lemmata 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Here we formulate a different version of G˚arding’s inequality compared to the case d = 2 considered in [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The sesquilinear form aN (i) is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant C > 0 independent of N such that |aN(w, v)| ≤ C∥w∥V ∥v∥V for all w, v ∈ V, and (ii) satisfies a G˚arding’s inequality in the form Re aN(v, v) ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (i) If the proof of [MS10, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4a)] is carried out with finitely many terms of the expansion of Tκ only, the statement follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Alternatively, Lemma 23 with s = 0 can also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (ii) As in the proof of Lemma 9, the definitions of aN and the wavenumber dependent norm yield Re aN(v, v) = ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR − Re (Tκ,Nv, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence it remains to estimate the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the case d = 2, we have (see (39)) Tκ,Nv(x) := 1 R � |n|≤N Zn(κR)vn(R)Yn(ˆx), x = Rˆx ∈ SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then, using the L2(S1)-orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], we get −(Tκ,Nv, v)SR = − 1 R � |n|≤N Zn(κR)(vn(R)Yn, vn(R)Yn)SR = − 1 R � |n|≤N Zn(κR)|vn(R)|2(Yn, Yn)SR = − � |n|≤N Zn(κR)|vn(R)|2(Yn, Yn)S1 = − � |n|≤N Zn(κR)|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 22 Resonant compactly supported nonlinearities January 30, 2023 Hence, by Lemma 4, − Re (Tκ,Nv, v)SR = � |n|≤N (− Re Zn(κR)) � �� � ≥1/2 |vn(R)|2 + (− Re Z0(κR)) � �� � >0 |v0(R)|2 ≥ 1 2 � |n|≤N |vn(R)|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The case d = 3 can be treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From Tκ,Nv(x) = 1 R N � n=0 � |m|≤n zn(κR)vm n (R)Y m n (ˆx) (see (40)), we immediately obtain, using the L2(S1)-orthonormality of the spherical harmon- ics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] that −(Tκ,Nv, v)SR = − 1 R N � n=0 � |m|≤n zn(κR)(vm n (R)Y m n , vm n (R)Y m n )SR = − 1 R N � n=0 � |m|≤n zn(κR)|vm n (R)|2(Y m n , Y m n )SR = −R N � n=0 � |m|≤n zn(κR)|vm n (R)|2(Y m n , Y m n )S1 = −R N � n=0 � |m|≤n zn(κR)|vm n (R)|2, and Lemma 4 implies − Re (Tκ,Nv, v)SR = R N � n=0 � |m|≤n (− Re zn(κR)) � �� � ≥1 |vm n (R)|2 ≥ R N � n=0 � |m|≤n |vm n (R)|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In both cases we obtain the same G˚arding’s inequality as in the original (untruncated) problem Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The next result is the variational version of the truncation error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It closely follows the lines of the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3], where an estimate of ∥(Tκ − Tκ,N)v∥s−1/2,2,SR, s ∈ R, was proved in the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For given w, v ∈ H1/2(SR) it holds that ���((Tκ − Tκ,N)w, v)SR ��� ≤ c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR, where c(N, w, v) ≥ 0 and limN→∞ c(N, w, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 23 Resonant compactly supported nonlinearities January 30, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the two-dimensional situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w(x) = w(Rˆx) = � |n|∈N0 wn(R)Yn(ˆx), v(x) = v(Rˆx) = � |k|∈N0 vk(R)Yk(ˆx), x ∈ SR, (42) be series representations of w|SR, v|SR with the Fourier coefficients wn(R) = (w(R·), Yn)S1 = � S1 w(Rˆx)Yn(ˆx)ds(ˆx), vk(R) = (v(R·), Yk)S1 = � S1 v(Rˆx)Yk(ˆx)ds(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6]: ∥v∥2 s,2,SR := R � n∈Z (1 + n2)s|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (43) Then, by (39), the orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] and (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ���((Tκ − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR ��� = 1 R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N � Zn(κR)wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N Zn(κR) (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = ������ � |n|>N Zn(κR)wn(R)vn(R) ������ = ������ � |n|>N Zn(κR) (1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) ������ ≤ max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� � |n|>N ��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) �� ≤ max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 where ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)2 := � |n|>N(1 + n2)1/2|wn(R)|2 � |n|∈N0(1 + n2)1/2|wn(R)|2 � |n|>N(1 + n2)1/2|vn(R)|2 � |n|∈N0(1 + n2)1/2|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The coefficient ˜c(N, w, v) tends to zero for N → ∞ thanks to (43), (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='. Corollary 5 implies the estimate 1 1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, |n| ∈ N0, hence we can set c(N, w, v) := ˜c(N, w, v) R max{|Z0(κR)|, (1 + |κR|2)1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The investigation of the case d = 3 runs similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w(x) = w(Rˆx) = � n∈N0 � |m|≤n wm n (R)Y m n (ˆx), v(x) = v(Rˆx) = � k∈N0 � |l|≤k vl k(R)Y l k(ˆx), x ∈ SR, (44) be series representations of w|SR, v|SR with the Fourier coefficients wm n (R) = (w(R·), Y m n )S1 = � S1 w(Rˆx)Y m n (ˆx)ds(ˆx), vl k(R) = (v(R·), Y l k)S1 = � S1 v(Rˆx)Y l k(ˆx)ds(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6]: ∥v∥2 s,2,SR := R2 � n∈N0 � |m|≤n (1 + n2)s|vm n (R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (45) Then, by (40), the orthonormality of the spherical harmonics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ���((Tκ − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR ��� = 1 R ������ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k>N � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k � zn(κR)wm n (R)Y m n (R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k(R−1·) � SR ������ = R ������ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k>N � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k zn(κR) � wm n (R)Y m n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k) � S1 ������ = R ������ � n>N � |m|≤n zn(κR)wm n (R)vm n (R) ������ = R ������ � n>N � |m|≤n zn(κR) (1 + n2)1/2(1 + n2)1/4wm n (R)(1 + n2)1/4vm n (R) ������ 25 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ R max n>N ���� zn(κR) (1 + n2)1/2 ���� � n>N � |m|≤n ��(1 + n2)1/4wm n (R)(1 + n2)1/4vm n (R) �� ≤ R max n>N ���� zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed� n>N � |m|≤n (1 + n2)1/2 |wm n (R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed� n>N � |m|≤n (1 + n2)1/2 |vm n (R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max n>N ���� zn(κR) (1 + n2)1/2 ���� ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' where ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)2 := � n>N � |m|≤n(1 + n2)1/2 |wm n (R)|2 � |n|∈N0 � |m|≤n(1 + n2)1/2 |wm n (R)|2 � n>N � |m|≤n(1 + n2)1/2 |vm n (R)|2 � |n|∈N0 � |m|≤n(1 + n2)1/2 |vm n (R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Thanks to Corollary 5 we can define c(N, w, v) := ˜c(N, w, v) R � 2 + |κR|2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For s ∈ [0, 1/2) and w ∈ H1−s(BR \\ Ω), v ∈ H1+s(BR \\ Ω) it holds that |(Tκ,Nw, v)SR| ≤ Cbl∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω, where the constant Cbl ≥ 0 does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the two-dimensional situation as in the proof of Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If w, v have the representations (42), then, by (39), the orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] and (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |(Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR| = 1 R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|≤N � Zn(κR)wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|≤N Zn(κR) (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = ������ � |n|≤N Zn(κR)wn(R)vn(R) ������ = ������ � |n|≤N Zn(κR) (1 + n2)1/2(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) ������ ≤ max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� � |n|≤N ��(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) �� 26 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed � |n|≤N (1 + n2)1/2−s |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|≤N (1 + n2)1/2+s |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� ∥w∥1/2−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2+s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5 implies the estimate 1 1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, |n| ∈ N0, hence |(Tκ,Nw, v)SR| ≤ 1 R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (46) By the trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='38], we finally arrive at |(Tκ,Nw, v)SR| ≤ C2 tr R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The investigation of the case d = 3 runs similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w, v have the representations (44), then, by (40), the orthonormality of the spherical harmonics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |(Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR| = 1 R ������ N � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k=0 � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k � zn(κR)wm n (R)Y m n (R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k(R−1·) � SR ������ = R ������ N � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k=0 � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k zn(κR) � wm n (R)Y m n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k(R)Y l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='= R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='= R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': 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+page_content='n∈N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='zn(κR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='��(1 + n2)(1/2−s)/2wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R)(1 + n2)(1/2+s)/2vm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='≤ R max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n∈N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='zn(κR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2−s |wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2+s |vm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Resonant compactly supported nonlinearities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ 1 R max n∈N0 ���� zn(κR) (1 + n2)1/2 ���� ∥w∥1/2−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2+s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5 yields |(Tκ,Nw, v)SR| ≤ 1 R � 2 + |κR|2�1/2 ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (47) By the trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='38], we finally arrive at |(Tκ,Nw, v)SR| ≤ C2 tr R � 2 + |κR|2�1/2 ∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, given an antilinear continuous functional ℓ : V → C, there exists a constant N∗ > 0 such that for N ≥ N∗ the problem Find uN ∈ V such that aN(uN, v) = ℓ(v) for all v ∈ V (48) is uniquely solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we show that the problem (48) has at most one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start as in the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5] and argue by contradiction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we suppose the following: ∀N∗ ∈ N ∃N = N(N∗) ≥ N∗ and uN = uN(N∗) ∈ V such that aN(uN, v) = 0 for all v ∈ V and ∥uN∥V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (49) However, the subsequent discussion differs significantly from the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We apply an argument the idea of which goes back to Schatz [Sch74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we assume there exists a solution uN ∈ V of (48) and derive an a priori estimate of the error ∥u − uN∥V , where u ∈ V is the solution of (29), see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since aN satisfies a G˚arding’s inequality (Lemma 21(ii)), we have, making use of (28), C2 −∥u − uN∥2 V − 2κ2∥u − uN∥2 0,2,BR ≤ Re aN(u − uN, u − uN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since aN(u − uN, v) = aN(u, v) − aN(uN, v) = a(u, v) � �� � =ℓ(v) +aN(u, v) − a(u, v) − aN(uN, v) � �� � =ℓ(v) = ((Tκ − Tκ,N)u, v)SR , we obtain C2 −∥u − uN∥2 V − 2κ2∥u − uN∥2 0,2,BR ≤ η1∥u − uN∥V (50) with η1 := sup v∈V Re ((Tκ − Tκ,N)u, v)SR ∥v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we consider the following auxiliary adjoint problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 43]): 28 Resonant compactly supported nonlinearities January 30, 2023 Find wN ∈ V such that a(v, wN) = (v, u − uN)BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (51) Since A is a Fredholm operator (see the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10), the adjoint problem possesses a unique solution wN ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then ∥u − uN∥2 0,2,SR = a(u − uN, wN) = a(u, wN) − a(uN, wN) = a(u, wN) − aN(uN, wN) � �� � =ℓ(wN)−ℓ(wN)=0 +aN(uN, wN) − a(uN, wN) = ((Tκ − Tκ,N)uN, wN)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, this relation shows that ((Tκ − Tκ,N)uN, wN)SR is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' With η2 := sup v∈V ((Tκ − Tκ,N)uN, v)SR ∥v∥V we obtain ∥u − uN∥2 0,2,BR ≤ η2∥wN∥V ≤ η2C−1 − C(R, κ)∥u − uN∥V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The continuous embedding V ⊂ V ∗ yields ∥u − uN∥2 0,2,BR ≤ η2C−1 − C(R, κ)Cemb∥u − uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Applying this estimate in (50), we get C2 −∥u − uN∥2 V − 2κ2η2C−1 − C(R, κ)Cemb∥u − uN∥V ≤ η1∥u − uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now, if ∥u − uN∥V ̸= 0, we finally arrive at C2 −∥u − uN∥V ≤ η1 + 2κ2η2C−1 − C(R, κ)Cemb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (52) Clearly this inequality is true also for ∥u − uN∥V = 0 so that we can remove this interim assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Thanks to Lemma 22 we have that ���((Tκ − Tκ,N)u, v)SR ��� ≤ c(N, u, v)∥u∥1/2,2,SR∥v∥1/2,2,SR ≤ c(N, u, c)C2 tr∥u∥V ∥v∥V , hence η1 ≤ c+(N, u)C2 tr∥u∥V with c+(N, u) := sup v∈V c(N, u, v), (53) where limN→∞ c+(N, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that, as can be seen from the proof of Lemma 22, the second fractional factor in the representation of ˜c(N, w, v) can be estimated from above by one without losing the limit behaviour for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Consequently, η1 can be made arbitrarily small provided N is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In order to estimate η2 we cannot apply Lemma 22 directly since the second argument in the factor c(N, uN, v) depends on N, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Therefore we give a more direct estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 29 Resonant compactly supported nonlinearities January 30, 2023 Namely, let v ∈ V have the representation (42) or (44), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then we define VN|SR := � span|n|≤N{Yn(R−1·)}, d = 2, spann=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N,|m|≤n{Y m n (R−1·)}, d = 3, and introduce an orthogonal projector PN : V |SR → VN|SR : v �→ PNv := �� |n|≤N vn(R)Yn(R−1·), d = 2, �N n=0 � |m|≤n vm n (R)Y m n (R−1·), d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then it holds that VN|SR ⊂ ker(TκPN − Tκ,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed, if d = 2 and v ∈ VN|SR, then PNv = v = � |n|≤N vn(R)Yn(R−1·) and TκPNv = Tκv = 1 R � |n|≤N Zn(κR)vn(R)Yn(R−1·) = Tκ,Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' An analogous argument applies in the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we return to the estimate of η2 and write, for uN ∈ V , (Tκ − Tκ,N)uN = (Tκ − TκPN)uN + (TκPN − Tκ,N)uN = Tκ(id −PN)uN, where we have used the above property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The advantage of this approach is that we can apply a wellknown estimate of the projection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The proof of this estimate runs similarly to the proof of Lemma 22 but only without the coefficients Zn or zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' respectively: ��((id −PN)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR �� = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N � wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = R ������ � |n|>N wn(R)vn(R) ������ = R ������ � |n|>N 1 (1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) ������ ≤ max |n|>N R (1 + n2)1/2 � |n|>N ��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) �� ≤ R (1 + N2)1/2 \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 (1 + N2)1/2∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 30 Resonant compactly supported nonlinearities January 30, 2023 The same estimate holds true for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then we get, by Remark 3 (or Lemma 23), ���((Tκ − Tκ,N)uN, v)SR ��� = ��(Tκ(id −PN)uN, v)SR �� ≤ Cκ (1 + N2)1/2∥uN∥1/2,2,SR∥v∥1/2,2,SR ≤ CC2 trκ (1 + N2)1/2∥uN∥V ∥v∥V , thus η2 ≤ CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Using this estimate and (53) in (52), we obtain C2 −∥u − uN∥V ≤ c+(N, u)C2 tr∥u∥V + 2κ2C−1 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (54) Now we appply this estimate to the solutions uN of the homogeneous truncated problems in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10, the homogeneous linear interior problem (29) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ℓ = 0) has the solution u = 0, and the above estimate implies C2 −∥uN∥V ≤ 2κ2C−1 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V , which is a contradiction to ∥uN∥V = 1 for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 allows an analogous conclusion as in Lemma 11 that the truncated bilinear form aN satisfies an inf-sup condition, such a conclusion is not fully satisfactory since the question remains whether and how the inf-sup constant depends on N or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, at least for sufficiently large N, a positive answer can given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, there exists a number N∗ ∈ N such that βN∗(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |aN(w, v)| ∥w∥V,κ∥v∥V,κ > 0 is independent of N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the proof a formula is given that expresses βN∗(R, κ) in terms of β(R, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We return to the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 and mention that the estimate (54) is valid for solutions u, uN of the general linear problems (29) (or, equally, (31)) and (48), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By the triangle inequality, ∥uN∥V ≤ ∥u∥V + ∥u − uN∥V ≤ ∥u∥V + c+(N, u)C−2 − C2 tr∥u∥V + 2κ2C−3 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 31 Resonant compactly supported nonlinearities January 30, 2023 If N∗ is sufficiently large such that κ2C−3 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2 ≤ 1 4 and c+(N, u)C−2 − C2 tr ≤ 1 for all N ≥ N∗, then, by Lemma 11, ∥uN∥V ≤ 4∥u∥V ≤ 4 C− ∥u∥V,κ ≤ ∥ℓ∥V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' That is, the sesquilinear form aN satisfies an inf-sup condition βN∗(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |aN(w, v)| ∥w∥V,κ∥v∥V,κ > 0 with βN∗(R, κ) := C−β(R, κ) 4C+ independent of N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Analogously to (30) we introduce the truncated linear operator AN : V → V ∗ by ANw(v) := aN(w, v) for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By Lemma 21, AN is a bounded operator, and Lemma 25 implies that AN has a bounded inverse: ∥w∥V,κ ≤ βN∗(R, κ)−1∥ANw∥∗ for all w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore, we define a nonlinear operator FN : V → V ∗ by FN(w)(v) := ℓcontr(w) + ℓsrc(w) + ℓinc N for all w ∈ V, where ⟨ℓinc N , v⟩ := (ˆx · ∇uinc − Tκ,Nuinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The problem (41) is then equivalent to the operator equation ANu = FN(u) in V ∗, and further to the fixed-point problem u = A−1 N FN(u) in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (55) Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, let the functions c and f generate locally Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore let uinc ∈ H1 loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If there exist numbers ̺ > 0 and LF ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ)) (where N∗ and βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ) are from Lemma 25) such that the following two conditions κ2 [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wf∥V ) � + Ctr∥ˆx · ∇uinc − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nuinc∥−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR ≤ ̺βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ2 [∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω̺ + ∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ LF are satisfied for all w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v ∈ Kcl ̺ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' then the problem (35) has a unique solution uN ∈ Kcl ̺ for all N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Analogously to the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 32 Resonant compactly supported nonlinearities January 30, 2023 7 Conclusion A mathematical model together with an investigation of existence and uniqueness of its solution for radiation and propagation effects on compactly supported cubic nonlinearities is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The full-space problem is reduced to an equivalent truncated local problem, whereby in particular the dependence of the solution on the truncation parameter (with regard to stability and errors) is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The results form the basis for the use of numerical methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', FEM, for the approximate solution of the original problem with controllable accuracy.' metadata={'source': 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