Generating customized field concentration via virtue surface transmission resonance
Yueguang Hu Hongyu Liu Xianchao Wang Deyue Zhang



§ ABSTRACT 
In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance.
The purpose of this paper is twofold.
Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field  to generate strong localized field concentration  at any specified places around the inclusion.  The aforementioned customized field concentration is crucially reliant on the peculiar  spectral and geometric patterns of certain transmission eigenfunctions.
Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying  strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.




Doping the spin-polarized Graphene minicone on Ni(111)
    Federico Bisti
    ^1 Université Paris-Saclay, CEA, Service de Thermo-hydraulique

et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France

^2 Laboratoire de Physique de la Matière Condensée, CNRS,
École Polytechnique, 
Institut Polytechnique de Paris, 91120 Palaiseau, France
=======================================================================================================================================================================================================================================================================



Keywords:   field concentration, transmission resonance,  surface localization, Herglotz approximation, oscillating behaviors

2010 Mathematics Subject Classification:  35P25, 35R30




§ INTRODUCTION




 §.§ Statement of main results

In this paper, we are concerned with artificially generating field concentration via surface transmission resonance. To begin with, we present the mathematical formulation of the transmission problem and discuss the major findings mainly focusing on the mathematics.

We consider a time-harmonic wave scattering problem arising from  an incident field u^i and a bounded inhomogeneous medium inclusion (Ω; σ,τ) in ^N,   N=2,  3, which is the mathematical model of our study.
Here, Ω denotes the  support of the inhomogeneity of the medium  with a connected complement in ^N and   σ,τ are the related material parameters. By choosing appropriate physical units, we may assume that the material parameters by normalization are set as

    σ̃() = χ_^N\Ω+σ()χ_Ωτ̃() = χ_^N\Ω+τ()χ_Ω∈^N ,

where σ() and τ() are two  L^∞ functions in Ω.  χ_A() is the indicator function such that  χ_A () =1 if ∈ A and χ_A () =0 otherwise.
Let u^i be an entire solution to homogeneous Helmholtz equation

    Δ u^i(𝐱) +k^2u^i(𝐱) =0  𝐱∈^N,

where k ∈_+ represents the (normalized) wavenumber of wave propagation. The incident field u^i impinging on the inclusion Ω gives rise to the scattered field u^s. Let u:= u^i +u^s denotes the total field.
The forward scattering problem is described by the following Helmholtz system,


    ∇·(σ̃∇ u )  + k^2 τ̃ u = 0    in ^N, 
    
    u = u^i +u^s     in ^N, 
    lim_r →∞ r^N-1/2(∂_r u^s-  k  u^s) =  0    r= ||,

where i = √(-1) is the imaginary unit and ∂_r u = ·∇ u with :=  / || ∈𝕊^N-1.
The last limit in (<ref>) characterizes the outgoing nature of the scattered  field,  known as the Sommerfeld radiation condition.
We also allow the case of absorption, represented by the material parameters
with nonzero imaginary parts. In this paper, we assume

    σ() ⩽ 0 τ() ⩾ 0 σ () ⩾γ_0 ∈Ω,

for some positive constant γ_0.
Then the system (<ref>) is well-posed,  and in particular, there exists a unique solution u ∈ H_loc^2(^N) <cit.>. Furthermore, the total field  has the following asymptotic expansion:

    u() = u^i () + e^ kr/r^(N-1)/2u_∞() +𝒪(1/r^(N+1)/2)    as   r → +∞,

which holds uniformly for all directions . Here the complex-value function u_∞, defined on the unit sphere 𝕊^N-1, is known as the far field  pattern and encodes the scattering information
caused by the perturbation of the incident field u^i due to the scatterer (Ω;σ,τ).

In this paper, we are concerned with artificially generating strong field concentration in a customized way. Specifically, we show that by properly choosing an incident field u^i, one can make the degree of field concentration, namely ∇ u_L^∞, to be arbitrarily
large at a customized exterior neighborhood around ∂Ω. The main result is presented as follows.


Consider the scattering problem (<ref>) with a C^1,1 inclusion Ω in ^N,N=2,3. We denote by Γ⊂∂Ω any subset of the boundary ∂Ω. Then  the gradient of the total field ∇ u can blow up in the following sense: for any given large number M>0, there exists an incident field u^i impinging on the inclusion Ω such that

    ∇ u_L^∞ (Γ_e(Ω,ϵ))⩾ M(u^i),

where M is only dependent of the incident field u^i and Γ_e(Ω,ϵ)= {x | dist(x,Γ) ⩽ϵ, x ∉Ω} is the exterior of the inclusion Ω along Γ for any given parameter ϵ = o(1).


The primary idea for generating field concentration involves introducing a virtual field concentration generator near the inhomogeneous medium inclusion (see Figure <ref>).
Subsequently, we can design a tailored incident field to activate such a generator and achieve the desired field concentration.
A field concentration generator is referred to as a transmission eigenmode ball where the wave field tends to concentrate around the boundary.
As the activated generator approaches to the inclusion, it produces a significant potential difference in the vicinity around the inclusion that lead to the blowup of gradient of the underlying wave field.
It should be noted that the construction of the incident field u^i is dependent on the distance parameter ϵ and the geometric location of the boundary part Γ. Therefore, the dependence of M on the incident field u^i also signifies its dependence on ϵ and Γ.
Furthermore, there are no restrictive requirements regarding the occurrence locations of field concentration. In fact, we can achieve field concentration in any specified location (even multiple locations) around the inclusion by introducing field concentration generators over there.

To enhance the clarity of the idea of generating field concentration, we outline the main proof of Theorem <ref> in this paper.
Firstly, we establish the existence of a sequence of specific transmission eigenfunctions within a radial domain for a given wavenumber (refer to Lemma <ref>). These transmission eigenfunctions are dependent on the construction of material parameters and as surface waves, they exhibit strong surface-localized properties (refer to Theorem <ref>). These properties satisfy the requirements for generating field concentration.
Secondly, the transmission eigenvalue problem (<ref>)  possesses a distinctive spectral pattern as a result of the topological structure of the defined domain (refer to Proposition <ref>). Based on this spectral pattern, we can construct the corresponding transmission eigenfunctions.
Thirdly, there exists a Herglotz wave function that can serve as a desirable approximation to the transmission eigenfunction. This Herglotz wave function satisfies the entire Helmholtz equation (<ref>) and can be assigned as the incident field (refer to Lemma <ref>).
Finally, by utilizing the nearly vanishing scattering property in the presence of transmission eigenfunctions (refer to Lemma <ref>), we can rigorously prove that the gradient of the total field would blow up at specified locations around the inhomogeneous medium inclusion.






As mentioned in the abstract, our study critically relies on certain geometric properties of the so-called transmission eigenfunctions, which were recently revealed in <cit.>. The transmission eigenvalue problem arises in the study of invisibility/transparency in continuum mechanics and has a very colorful history. For further discussion on this topic, we refer the readers to <cit.>.
In this paper, we reexamine the transmission eigenvalue problem (<ref>) from the standpoint of material construction.
We have proved that for a given wavenumber k, there exists a sequence of transmission eigenfunctions which exhibit strong surface resonance behaviors. Specially, these transmission eigenfunctions together with their gradients tend to localize/concentrate on the boundary surface of the underlying inhomogeneous inclusion. Moreover, their oscillating frequencies along the boundary are significantly higher than the excitation frequency. The main result is presented as follows.




Consider the transmission eigenvalue problem (<ref>) in a radial domain B_r_0 centering the origin with radius r_0 in ^N, N=2,3 and a given constant wavenumber k. Then there exists a sequence of transmission eigenfucntions (v_m,w_m)_m ∈  depending on the material parameters (σ_m,τ_m)_m ∈ such that k  is the transmission eigenvalue and the transmission eigenfunctions and their gradients are surface-localized, i.e.

    lim_m→∞ψ_m_L^2(B_ξ )/ψ_m_L^2(B_r_0) =0 lim_m→∞∇ψ_m_L^2(B_ξ)/∇ψ_m_L^2(B_r_0) = 0  ,   ψ_m =w_m, v_m ,

where  B_ξ is defined as

    B_ξ= {x ∈ B_r_0, dist(x,∂ B_r_0)⩾ξ r_0,ξ∈ (0,1)}.

We further suppose v_m is L^2(B_r_0)-normalised, i.e. v_m_L^2(B_r_0) =1, then

    lim_m→∞∇ψ_m_L^∞(Σ_ξ )/k = ∞ ,

where Σ_ξ is defined as

    Σ_ξ =
    { (r,θ) | ξ r_0 < r < r_0, θ_1 < θ <θ_2}ξ∈ (0,1); θ_1,θ_2 ∈  [0,2π) , 
    {(r,θ,φ) | ξ r_0 < r < r_0,  0⩽θ⩽π ,φ_1 < φ <φ_2}ξ∈ (0,1); φ_1,φ_2 ∈ [0,2π).



B_ξ and Σ_ξ represent the interior and boundary domain inside the inclusion respectively. In theorem <ref>,
the first limit indicates that these transmission eigenfucntions are surface-localized eigenmodes.
The second limit indicates that their most oscillating behaviors tend to concentrate along the boundary for both transmission eigenfunctions.
In the vicinity along the boundary, The last limit indicates that the oscillating frequencies of transmission eigenfunctions are much larger than the natural frequency k, especially for sufficiently large order m.











 §.§ Background discussion and scientific novelties



In the field of fluid mechanics, a field concentration refers to a location within an object where the gradient of velocity potential or pressure is substantially higher than the surrounding region in the fluid field.
Excessively large velocity gradient can lead to serious consequences, including flow instability and structural damage.
Nevertheless, the destructive nature of field concentration can be harnessed for practical applications.
Proper utilization of field concentration offers engineers numerous possibilities to analyze and optimize the performance of fluid systems.
Therefore, comprehending field concentrations is fundamental in fluid mechanics and plays a vital role in designing and optimizing various engineering applications.


Field concentration can generally be utilized to perform destructive tasks in engineering.
In medical applications, such as treating cancers (e.g., prostate cancer, liver tumors) and non-cancerous conditions (e.g., uterine fibroids), a transducer focuses acoustic waves on specific points within the body to creat high-intensity regions where the acoustic energy concentrates and lead to thermal ablation of the target tissue. The concentrated acoustic field can generate localized energy concentration capable of destroying pathological tissues, including tumors.
Similarly, in the treatment of kidney stones and gallstones, lithotripsy involves generating a series of acoustic waves outside the body, concentrating acoustic energy at the location of the stone and render it fragile. Subsequently, field concentration can fragment kidney stones or gallstones into smaller pieces that can be naturally expelled by the body.
On the other hand, field concentration can also enhance the performance of fluid systems.
In the design of aircraft lifting surfaces, the low-pressure region on the upper surface of the airfoil and the corresponding pressure gradients are utilized to generate the required lift. Optimizing the distribution of field concentration can improve lift and reduce drag.
Nozzles and propulsion systems also exemplify this application. The design of rocket engines and jet engines relies on the extremely high velocity potential gradient and pressure gradient in the exhaust flow to generate thrust. Through meticulous nozzle system design, the velocity potential gradients can be optimized to enhance the propulsive efficiency.




As mentioned earlier, we are concerned with artificially creating strong field concentrations in a customized way.  The field concentration can occur at a customized place around a generic material inclusion or within the material inclusion if it possesses a certain specific geometric structure.
There are several novel and salient features of our study. First, the field concentration has been widely investigated in the literature, say e.g. <cit.> in the high-contrast composite material theory. It turns out that the material and geometric irregularities are crucial for the occurrence of field concentration. In fact, it is usually required that the material parameters of the inclusion  and those of background medium possess a certain asymptotically high contrast. Moreover, it usually occurs between two close-to-touching convex material inclusions, corresponding to the building blocks of the underlying composite material. The study is usually concerned with the static or quasi-static case, namely ω = 0 or w ·diam(Ω) ≪ 1 see <cit.> for discussion on the related developments in the literature. However, for our study, there are none of such restrictive requirements: there can be only one material inclusion with regular material parameters of generic geometric shape, and the field concentration can occur in the quasi-static regime or beyond the quasi-static regime. Second, a large field concentration may cause the failure of the material structure. Many preventive methods have been developed to counter the destructive effect of field concentration. To our best knowledge, there is little study on artificially and customarily generating strong field concentration within a material structure. As is well known, everything might have many different sides. The destructive nature of field concentration can be put into good use. For example, in the treatment for kidney stones and gallstones, a destructive failure of the  stone structure inside can make the treatment easier and safer and reduce the need for surgical intervention. Finally, we would like to mention that gradient estimates of solutions is a central topic in the theory of partial differential equations (PDEs); see e.g. <cit.>. Our study provides a good example that for wave equations, the gradient blowup phenomena may be even induced for (possibly) smooth coefficients due to frequency effect.

The rest of the paper is organized as follows.
Section 2 presents the spectral patterns of transmission resonance and provides the proof for the existence of a sequence of surface transmission resonant eigenfunctions associated with a given wavenumber. These eigenfunctions are utilized to construct transmission resonant balls, which facilitate the generation of strong field concentration.
Section 3 demonstrates the artificial generation of customized field concentration via the spectral and geometric patterns of transmission resonance. It is followed by the illustration of this phenomenon through numerical experiments in Section 4.
Section 5 focuses on the surface-oscillating behaviors of transmission resonance and concludes the complete proof of Theorem <ref>.






§ SURFACE TRANSMISSION RESONANCE

This section introduces the transmission eigenvalue problem along with its spectral patterns, and subsequently establishes the existence of a sequence of surface transmission resonant eigenfunctions.



 §.§ Transmission eigenvalue problem and its spectral patterns

The transmission eigenvalue problem is associated with  invisibility in  the wave scattering system (<ref>). When invisibility occurs, there is no perturbation outside caused by the inhomogeneous medium inclusion,  i.e., u^s≡ 0. This reduction leads to the following (interior) transmission eigenvalue problem:


    ∇·(σ∇ w )  +k^2τ w = 0    in  B, 
    Δ  v +k^2v = 0    in  B, 
     w =v σ∂ w/∂ν = ∂ v/∂ν   on ∂ B,

where ν∈𝒮^N-1 is the exterior unit normal to ∂ B and B is a bounded Lipschitz domain in ^N.
In order to make a distinction between the inhomogeneous medium inclusion introduced in (<ref>) and the definition domain  introduced in (<ref>), we introduce the symbol B to denote the definition domain for transmission eigenvalue problem. By slight abuse of notation, B_r(_0) represents a radial domain with radius r and center _0 in ^N. When _0 is the origin, it is omitted that should be evident from the context. Additionally, we introduce an auxiliary parameter to facilitate our analysis:

    n =√(τ/σ).

It is evident that  w = v = 0 forms  a pair of trivial solutions to (<ref>).
If the system (<ref>) admits a pair of nontrivial solutions (v,w), then k is referred to as a transmission eigenvalue, and (v,w) are the corresponding transmission eigenfunctions. The transmission eigenfunction in fact can be viewed as the restriction of the wave inside the inhomogeneous medium inclusion. The behaviors of transmission eigenfunction reflect the geometric properties of  the wave probing on the invisible inclusion. This can help to understand the mathematical nature of invisibility.
In this paper, we find out there exists a sequence of surface-localized transmission eigenfunctions (v_m,w_m)_m ∈. Here, the localization  means that there exists a sufficiently small number ε =o(1) such that

    ψ_L^2(B_ε)/ψ_L^2(B) = o(1) ,   ψ = w  and  v.

where B_ε:= {x ∈ B, dist(x,∂ B)⩾ε}.
If localization occurs, the corresponding transmission eigenfunctions are called surface-localized transmission eigenmodes.

Next we shall consider the spectral pattern of transmission eigenvalue problem.  Let Ω be composed of multiple simply-connected components as follows:

    (Ω;σ,τ) = ∪_j=1^L_0(Ω_j;σ_j,τ_j),

where Ω_j's are mutually disjoint and (Ω_j;σ_j,τ_j) are the restriction of (Ω;σ,τ) on Ω_j.

The following theorem states that the set of all transmission eigenvalues in Ω is in fact the union of the eigenvalues in each component Ω_j. Moreover, the transmission eigenfunctions in Ω are linear combinations of the eigenfunctions in each component Ω_j with the trivial eigenfunctions in other components.



Let σ [Ω;σ,τ] and σ[Ω_j;σ_j,τ_j] be the set of transmission eigenvalues of the system (<ref>) in (Ω;σ,τ) and  (Ω_j;σ_j,τ_j), 1⩽ j ⩽ L_0,  where (Ω;σ,τ) = ∪_j=1^L_0(Ω_j;σ_j,τ_j). Then

    σ [Ω;σ,τ] = ∪_j=1^L_0σ[Ω_j;σ_j,τ_j].



We first prove that

    ∪_j=1^L_0σ[Ω_j;σ_j,τ_j]⊂σ [Ω;σ,τ].

Without loss of generality, we consider k∈σ[Ω_1;σ_1,τ_1] with the associated transmission eigenfunctions w_1∈ H^1(Ω_1) and v_1∈ H^1(Ω_1), that is,

    ∇· (σ_1∇ w_1) +k^2 τ_1 w_1 = 0    in  Ω_1, 
    Δ v_1 +k^2v_1 = 0    in  Ω_1, 
    
    w_1 =v_1 σ_1∂ w_1/∂ν = ∂ v_1/∂ν   on  ∂Ω_1,

Next, it is trivially seen that w_j=v_j=0, j=2, 3,…, L_0, satisfy

    ∇· (σ_j∇ w_j) +k^2 τ_j w_j = 0     in  Ω_j, 
    Δ v_j +k^2v_j = 0    in  Ω_j, 
    
    w_j =v_j σ_j∂ w_j/∂ν = ∂ v_j/∂ν   on  ∂Ω_j,

Set

    w=    w_1χ_Ω_1+0·χ_Ω_2+⋯+0·χ_Ω_L_0∈ H^1(Ω),
    
    v=    v_1χ_Ω_1+0·χ_Ω_2+⋯+0·χ_Ω_L_0∈ H^1(Ω).

By virtue of (<ref>) and (<ref>) as well as using the fact that Ω_j are mutually disjoint, it is directly seen that w and v defined in (<ref>) are nontrivial solutions to (<ref>) with k=k_1. Hence, k_1∈σ [Ω;σ,τ], which readily proves (<ref>).

We proceed to prove that

    σ [Ω;σ,τ]⊂∪_j=1^L_0σ [Ω_j; σ_j,τ_j].

Let k∈σ [Ω;σ,τ] and w, v be the associated transmission eigenfunctions. Since w∈ H^1(Ω) and v∈ H^1(Ω) are non-identically zero and Ω_j's are mutually disjoint, there must exist an Ω_j_0, 1≤ j_0≤ L_0, such that

    w_j_0=w|_Ω_j_0∈ H^1(Ω_j_0),    v_j_0=v|_Ω_j_0∈ H^1(Ω_j_0),

are non-identically zero. Again, by using the facts that Ω_j's are mutually disjoint and w, v satisfy (<ref>), it can be seen that

    ∇· (σ_j_0∇ w_j_0) +k^2 τ_j_0 w_j_0 = 0     in  Ω_j_0, 
    Δ v_j_0 +k^2v_j_0 = 0    in  Ω_j_0, 
    
    w_j_0 =v_j_0σ_j_0∂  w_j_0/∂ν = ∂ v_j_0/∂ν   on  ∂Ω_j_0,

That is,

    k∈σ[Ω_j_0;σ_j_0,τ_j_0]⊂∪_j=1^L_0σ [Ω_j;σ_j,τ_j],

which proves (<ref>).

Finally, combining (<ref>) and (<ref>) readily yields (<ref>), thus completing the proof.





 §.§ The geometric patterns of transmission resonance

The transmission eigenvalue problem (<ref>) possesses a sequence of transmission eigenvalues  and surface-localized transmission eigenfunctions for the given material parameters (σ,τ) <cit.>. In this subsection, we reexamine the transmission eigenvalue problem for the given wavenumber k and  radial domain B_r_0. We prove that there exists a sequence of {n_m}_m ∈ such that k is the transmission eigenvalue and the corresponding  transmission eigenfunctions (v_m,w_m)_m ∈ are surface-localized. This allows us  to construct the field concentration generator with arbitrary size by considering the material parameter.
These eigenfunctions are in fact surface transmission resonant modes, accompanying strong surface-localization and manifesting highly-oscillatory patterns along with the energy blowup.
Since the verification  of surface-oscillating patterns  involves technical and tedious calculations, we defer the related proof to the section <ref>.




To begin with, we denote J_m(x) and j_m(x)  respectively denote the m-th order Bessel function and m-th order spherical Bessel function. Moreover, let j_m,s and j'_m,s respectively denote the s-th positive root of J_m(x) and J'_m(x) (arranged according to magnitude).


Consider the transmission eigenvalue problem (<ref>) and assume B_r_0 is a radial object with radius r_0 in ^N centering the origin. Given any wavenumber k>0, there exists a sequence of {n_m}_m∈ such that k is the transmission eigenvalue and the corresponding transmission eigenfunctions (v_m,w_m) are both surface-localized. In specific, let s_0 ∈ be a fixed positive integer,  the eigenvalues  are

    n_m ∈ (j_m,s_0/kr_0,j_m,s_0+1/kr_0)   in ^2  n_m ∈ (j_m+1/2,s_0/kr_0,j_m+1/2,s_0+1/kr_0)   in ^3,

and the corresponding eigenfunctions in ^2 are

    w_m() = α_mJ_m(kn_mr)e^ mθ v_m() = β_m J_m(kr)e^ mθ ,

where  = (rcosθ,rsinθ) ∈^2 denotes the polar coordinate. The corresponding eigenfunctions in ^3 are

    w_m^l() = α_m^l j_m(kn_mr)Y_m^l(θ,φ)  v_m^l() = β_m^l j_m(kr)Y_m^l(θ,φ)   -m ⩽  l ⩽ m ,

where = (rsinθcosφ,r sinθsinφ,rcosθ) ∈^3 denotes the spherical coordinate and Y_m^l are the spherical harmonics.



To ensure that (v_m,w_m) are the transmission eigenfunctions of (<ref>), the transmission boundary conditions on the last line in (<ref>) yield

    α_m J_m(knr_0) = β_m J_m(kr_0)   and  σ n α_m  J'_m(knr_0) = β_m J'_m(kr_0).

So n must be the root of the following equation:

    f_m(n) = J'_m(kr_0)J_m(knr_0) -σ nJ_m(kr_0)J'_m(knr_0).

For a given integral s_0 ∈, it is obvious that J_m(knr_0) = 0 when n = j_m,s_0/(kr_0) and j_m,s_0+1/(kr_0) respectively.
From the fact that the zeros of J_m(x) and J'_m(x) are interlaced, it holds that

    f_m(j_m,s_0/kr_0) · f_m(j_m,s_0+1/kr_0) = σ^2 j_m,s_0j_m,s_0+1/k^2r_0^2 J^2_m(kr_0)J'_m(j_m,s_0)J'_m(j_m,s_0+1) < 0.

It readily implies by Rolle's theorem that there exists at least one zero denoted by n_m in (j_m,s_0/(kr_0),[2]j_m,s_0+1/(kr_0)) satisfying f_m(n_m) =0. Then the transmission eigenvalue is given by

    n_m ∈(j_m,s_0/kr_0 ,j_m,s_0+1/kr_0),

and  (v_m,w_m) are the corresponding transmission functions.

 The surface-localizations of (v_m,w_m) mainly depend on the properties of Bessel functions J_m and spherical Bessel functions j_m. From the orthogonality of {e^ mθ}_m ∈ in the unit circle, we can obtain

    v_m^2_L^2(B_r_0) =β_m^2 ∫_0^r_0  J^2_m(kr)r r =  β_m^2/k^2∫_0^kr_0 J^2_m(r)r r.

Without loss of generality, we assume kr_0 is always less than m and  this assumption always holds when m is sufficiently large. We define a monotonously increasing and convex function

    f(r) = J^2_m(r)r  r ∈ (0,kr_0).

In fact,  straight calculations yield

    f'(r)    = J_m(r)(2J'_m(r)r+J_m(r)) >0 ,
    
    f”(r)    = 2J'^2_m(r)r + 2J'_m(r)J_m(r) + 2r^-1(m^2-r^2)J_m^2(r) >0.


The convex property means that the integral (<ref>) is bigger than the area of the triangle under the tangent of f(kr_0) with the x-axis, namely,

    ∫_0^kr_0 J^2_m(r)r r⩾1/2J_m^3(kr_0)(kr_0)^2/J_m(kr_0)+2kr_0J'_m(kr_0).

Since f(r) is  monotonously increasing, it is obvious that

    ∫_0^kξ r_0 J^2_m(r)r r⩽1/2 J^2_m(kξ r_0)(kξ r_0)^2 ξ∈ (0,1) .

Combining the above two estimates, we further obtain

    v_m_L^2(B_ξ)^2/v_m_L^2(B_r_0)^2   ⩽J_m^2(kξ r_0)(kξ r_0)^2/J_m^3(kr_0)(kr_0)^2/(J_m(kr_0)+2kr_0J'_m(kr_0))
       = (J_m(kξ r_0)/J_m(kr_0))^2 ξ^2(1 + 2kr_0J'_m(kr_0)/J_m(kr_0)).

When m is sufficiently large, it holds that

    J'_m(kr_0)/J_m(kr_0) = m/kr_0 - J_m+1(kr_0)/J_m(kr_0) = m/kr_0(1 - k^2r_0^2/2m(m+1) + 𝒪(m^-3)).

where we have used the asymptotic expansions of J_m for larger order,

    J_m(x) = x^m/2^m Γ(m+1)(1 - x^2/4(m+1) + 𝒪(m^-2) ),

and Γ(x) is the Gamma function. By using the asymptotic expansions (<ref>) again, we can get

    J_m(kξ r_0)/J_m(kr_0) = (kξ r_0)^m(1 - (kξ r_0)^2/4(m+1) + 𝒪(m^-2) )/(kr_0)^m(1 - (kr_0)^2/4(m+1) + 𝒪(m^-2) )∼ξ^m.

Finally, we insert (<ref>) and (<ref>) in  (<ref>) and get

    v_m_L^2(B_ξ)^2/v_m_L^2(B_r_0)^2⩽ C(k,r_0)ξ^2m+2(1+2m) → 0    as    m →∞.

We have proved the surface-localization of {v_m}_m∈. Similarly,
the L^2-norm of w_m is

    w_m^2_L^2(B_r_0) =α_m^2 ∫_0^r_0  J^2_m(kn_mr)r r =  β_m^2/k^2n_m^2∫_0^kn_mr_0 J^2_m(r)r r.

Since k n_m r_0 ∈ (j_m,s_0,j_m,s_0+1) and the zeros of Bessel functions have the following asymptotic expansions,

    j_m,s_0 = m + b_s_0m^1/3 + 𝒪(m^-1/3),

where b_s_0 is some positive constant only depending on the fixed integer s_0 ∈, then it holds that

    kn_mr_0 > j_m,1 >j'_m,1>m > kn_m τ r_0.

By following a similar argument as (<ref>), we can readily get

    w_m_L^2(B_ξ)^2/w_m_L^2(B_r_0)^2   =  ∫_0^kn_mξ r_0 J^2_m(r)r r/∫_0^kn_mr_0 J^2_m(r)r r⩽∫_0^kn_mξ r_0 J^2_m(r)r r/∫_0^m J^2_m(r)r r
       ⩽1/2J_m^2(kn_mξ r_0)(kn_mξ r_0)^2/1/2J_m^3(m)m^2/(J_m(m)+2mJ'_m(m))
       = (J_m(kn_mξ r_0)/J_m(m))^2 (kn_mξ r_0/m)^2(1 + 2mJ'_m(m)/J_m(m)).

On the one hand, kn_mξ r_0 <m and

    1 + 2mJ'_m(m)/J_m(m)   = 1 +2m (1 - J_m+1(m)/J_m(m)) = 1 + m( m+2/m+1 - m/ (m+1)J_m+2(m)/J_m(m)) 
       ⩽ 1 +  m(m+2)/m+1⩽ m+2.

where we have used the recurrence formula J_m(x) +J_m+2(x) = 2(m+1)J_m+1(x)/x. On the other hand,  from <cit.>, one can get

    J_m(kn_mξ r_0)/J_m(m)∼ C(k,ξ) Ai(m^2/3ζ_x) ∼ C(k,ξ)  e^-2/3ζ_x^3/2mm^-1/6,

where

    ζ_x=(3/2(ln(1+√(1-x^2)/x)-√(1-x^2)))^2/3 x =k n_mξ r_0/m ∼ξ ,

and we have used the  asymptotic formula of Airy function for sufficiently x,

    Ai(x) = e^-2/3x^3/2/2π^1/2x^1/4(1 - 5/48x^-3/2 +𝒪(x^-3) ).

Finally, we can further obtain

    w_m_L^2(B_ξ)^2/w_m_L^2(B_r_0)^2   ⩽  C(k,ξ) e^-2/3ζ_x^3/2m(m+2)m^-1/6→ 0   as m →∞.

Therefore, we have proved the surface-localization of {w_m}_m∈.



The proof in ^3 is similar with that in ^2.
First, the transmission boundary conditions for (v_m^l,w_m^l) defined on (<ref>) yield

    f_m(n) = j'_m(kr_0)j_m(knr_0) -σ nj_m(kr_0)j'_m(knr_0).

It is noted that the zeros of j_m(x) are the same as that of J_m+1/2(x). Taking the value of n=j_m+1/2,s_0/kr_0 and n =j_m+1/2,s_0+1/kr_0 respectively, we can get

    f_m( j_m+1/2,s_0/kr_0)f_m( j_m+1/2,s_0+1/kr_0) = σ^2  j_m+1/2,s_0j_m+1/2,s_0+1/k^2r_0^2j'_m(j_m+1/2,s_0)j'_m(j_m+1/2,s_0+1).

It holds from j_m(x)  =√(π /2x)J_m+1/2(x) that

    . j'_m(x)|_x = j_m+1/2,s_0 = .( √(π/2x) J_m+1/2(x))'|_x = j_m+1/2,s_0 = √(π/2j_m+1/2,s_0)J'_m+1/2(j_m+1/2,s_0).

Since  the zeros between J_m+1/2(x) and J'_m+1/2(x) are interlaced, it is clear  that

    f_m( j_m+1/2,s_0/kr_0)f_m( j_m+1/2,s_0+1/kr_0)< 0.

By the  Rolle's theorem again, there exists at least one zero also denoted by n_m satisfying f_m(n_m) = 0, namely,

    n_m ∈(j_m+1/2,s_0/kr_0 ,j_m+1/2,s_0+1/kr_0).

The surface-localization of (v_m,w_m) is obvious from the results in ^2. Indeed, the L^2-norm of (v_m,w_m) are given by

    v^l_m^2_L^2(B_r_0)   =(α_m^l)^2 ∫_0^r_0  j^2_m(kr)r^2 r =  (α_m^l)^2/k^2∫_0^kr_0 J^2_m+1/2(r)r r, 
    w^l_m^2_L^2(B_r_0)   =(β_m^l)^2 ∫_0^r_0  j^2_m(kn_mr)r^2 r =  (β_m^l)^2/k^2n_m^2∫_0^kn_mr_0 J^2_m+1/2(r)r r.

By substituting the order of Bessel function from m to m+1/2 and following the same argument as (<ref>) and (<ref>), we can easily obtain the surface-localized result for  (v_m,w_m).

















































§ ARTIFICIAL GENERATION OF CUSTOMIZED FIELD CONCENTRATIONS


In this section, we shall show how to artificially generate customized field concentration around the inclusion via surface transmission resonance.
To that end, we first introduce the Herglotz  approximation and prove the nearly vanishing property of scattered field in the presence of transmission eigenfunctions.











 §.§ Herglotz approximation

A Herglotz function is a function of the form

    v_g,k = H_kg () := ∫_𝕊^N-1e^ik·θ g(θ) θ∈^N,

where g(θ) ∈ L^2(𝕊^N-1) is called the Herglotz kernel.
It is clear that the Herglotz function is an entire function  to the Helmholtz equation (<ref>) and can be designated as the incident field.
For the purpose of generating field concentration, we shall use a customized Herglotz function to activate a field concentration generator around the inhomogeneous medium inclusion.
Here we introduce the following denseness result of the Herglotz function.
 <cit.>  
Suppose that Ω is of class C^α,1 , α∈∪{0} with a connected complement in ^N,N=2,3. Denote by ℋ be the space of all Herglotz functions and

    ℌ(Ω) := {u ∈ C^∞ (Ω) Δ u +k^2 u =0   in  Ω}

Then ℋ(Ω) is dense in ℌ(Ω)  ∩ H^α +1(Ω) with respect to the H^α+1 norm.





 §.§ The nearly vanishing property of  scattered field

We begin to prove the smallness of scattered field in the presence of transmission eigenfunction. This result is in fact a straight consequence of the
stability estimate for the the scattering system (<ref>).
To that end, we first truncate the scattering system (<ref>) into a bounded ball D_R with a sufficiently large radius R.

The scattering system (<ref>) is equivalent to the following truncated system:

    Δ u^s_1 + k^2 u^s_1 = 0     in  D_R \Ω, 
    ∇·(σ∇ u_1) + k^2 τ u_1 = 0     in Ω, 
    
    u_1 = u^s_1 + f σ∂_ν u_1 = ∂_ν u^s_1 + g    on ∂Ω, 
    ∂ u^s_1/∂ν = Λ u^s_1    on ∂ D_R ,

where  f = u^i|_∂Ω , g = ∂_ν u^i|_∂Ω and
Λ: H^3/2(∂Ω) → H^1/2(∂Ω) is the Dirichlet-to-Neumann map defined by Λ W := ∂ V/∂ν with V ∈ H_loc^2(^N \D_R) is the unique solution of

    Δ V + k^2 V = 0     in ^N \D_R, 
    
    V = W    on ∂ D_R , 
    lim_r →∞ r^N-1/2(∂_r V-ik V) =  0    r= ||.



Let (u,u^s) ∈ H^2(Ω) × H_loc^2(^N \Ω) be a solution of (<ref>). From the definition of Λ, the restriction of u^s in D_R \Ω is clearly a solution of (<ref>). Conversely,  let  (u_1,u^s_1) ∈ H^2(Ω) × H_loc^2(^N \Ω) be a solution of (<ref>). We shall prove that the extension of u^s_1 from D_R \Ω to outside of D_R satisfies the Sommerfeld radiating condition. From the Green's formula for Helmholtz equation, the representation of u^s_1 is

    u^s_1 =   ∫_∂Ω u^s_1(y) ∂Φ(x,y)/∂ν(y) - ∂ u^s_1/∂ν(y) Φ(x,y)s(y)
       - ∫_∂ D_R u^s_1(y) ∂Φ(x,y)/∂ν(y) - Λ u^s_1(y)  Φ(x,y)s(y),   x ∈ D_R \Ω,

where Φ(x,y) is the fundamental solution to Helmholtz equation in ^N. From the definition of Λ, u^s_1 is in fact the boundary data of the radiating solution V to (<ref>). Together with the fact that Φ is also a radiating solution to Helmholtz equation,  for x ∈ D_R \Ω, we can deduce that

    ∫_∂ D_R u^s_1(y) ∂Φ(x,y)/∂ν(y) - Λ u^s_1(y)  Φ(x,y)s(y)
       = lim_r →∞∫_∂ B_r V(y) (∂Φ(x,y)/∂ν(y) - k Φ(x,y) ) - ( ∂ V/∂ν(y) -  k V(y)) Φ(x,y)s(y)
        =0,


where we have used the boundedness of radiation solution and Sommerfeld radiation condition. Hence, u_1^s coincides with the radiating solution to the (<ref>) in the exterior of Ω. We get (u_1,u^s_1) is also a solution to (<ref>).



Consider the scattering system (<ref>) with a C^1,1 incluson (Ω;σ,τ) in ^N, N=2,3 and the wavenumber k∈_+.
We let (v,w) be a pair of transmission eigenfunction in  (Ω;σ,τ) associated with k.
If the incident field u^i impinging on (Ω;σ,τ) satisfies u^i-v_H^2(Ω)⩽ε,  then there exists a constant C(k,σ,τ) such that the scattered field u^s illuminated by u^i has u^s_H^2_loc(^N \Ω)⩽ C ε.


For Lemma <ref>, we can rewrite (<ref>) in the following form:

    Δ u^s + k^2 u^s = 0     in  D_R \Ω, 
    ∇·(σ∇ u) + k^2 τ u = 0     in Ω, 
    
    u = u^s + f σ∂_ν u = ∂_ν u^s + g    in ∂Ω, 
    ∂ u^s/∂ν = Λ u^s    on ∂ D_R ,

where f = u^i|_∂Ω and g = ∂_ν u^i|_∂Ω and Λ is defined in (<ref>). We first introduce the solution u_f being the unique solution of

    Δ u_f + k^2 u_f = 0     in  D_R \Ω, 
    
    u _f = f     on ∂Ω, 
    
    u _f = 0     on ∂ D_R.

Without loss of generality, we assume that  k^2 is not the Dirichlet eigenvalue of -Δ in D_R \Ω through a proper choice D_R.

By introducing the test function ψ∈ H^1(D_R), the equivalent variational formulation of (<ref>) is to find p ∈  H^1(D_R) such that

    ∫ _Ωσ∇ p  ∇ψ -k^2 τ p ψx+ ∫ _D_R\Ω∇ p ∇ψ -k^2  p ψx  - ∫_∂ D_RΛ p ψs
    
    =   ∫ _D_R\Ω  -∇ u_f  ∇ψ + k^2  u_f ψx + ∫_∂ D_R∂_νu_f  ψs + ∫_∂Ω(g - ∂_νu_f )ψs∀ψ∈ H^1(D_R).

By using Green's first theorem, it is easy to check that u:= p|_Ω and u^s := p|_ D_R \Ω -u_f  satisfies the scattering system (<ref>). On the other hand, we can also get the variational formulation (<ref>) by multiplying the test function to the first two equations in (<ref>).

In order to prove the regularity result of (<ref>), we first introduce a bounded operator Λ_0:H^1/2(∂ D_R) → H^-1/2(∂ D_R) defined in the Theorem 5.20 <cit.> and satisfies

    - ∫_∂ D_RΛ_0 ψψs⩾ C ψ^2_H^1/2(∂ D_R) ,

for some C>0. Further, the operator Λ -Λ_0 is a compact operator from H^1/2(∂ D_R) to H^-1/2(∂ D_R). Then  the variational formulation (<ref>) can be rewritten as

    a_1(p, ψ) + a_2(p, ψ) + ⟨ -(Λ-Λ_0)p,ψ⟩ = ℱ(ψ) ∀ψ∈ H^1(D_R),

where ⟨· ,·⟩ denotes the inner product in L^2(∂ D_R),  and the bilinear forms a_1 and a_2 together with the linear bounded functional f are defined by

    a_1(p, ψ)     := ∫ _Ωσ∇ p  ∇ψ +  k^2 pψx + ∫ _D_R\Ω∇ p  ∇ψ + k^2 pψx - ∫_∂ D_RΛ_0 p ψs, 
    
    a_2(p,ψ)    := -∫ _Ω k^2(1+τ)p ψx -∫ _Ω 2k^2 p ψx ,


and

    ℱ(ψ):=∫ _D_R\Ω  -∇ u_f  ∇ψ + k^2  u_f ψx + ∫_∂ D_R∂_νu_f  ψs + ∫_∂Ω(g - ∂_νu_f )ψs.

From the assumption of σ and boundedness of Λ_0, it is clear that

    a_1(p,ψ) ⩽ C p_H^1(D_R)ψ_H^1(D_R) a_1(p,p) ⩾ C p^2_H^1(D_R) ,

where C is some positive constant. The Lax-Milgram lemma  indicates there exists a unique linear bounded operator 𝒜 with a bounded inverse such that

    a_1(p,ψ) = (𝒜p,ψ) ∀   p,ψ∈ H^1(D_R).

By the Riesz representation theorem, there also exists a bounded operator B:H^1(D_R) → H^1(D_R) such that

    a_2(p,ψ) = (ℬp,ψ) ∀ p,ψ∈ H^1(D_R).

We claim that ℬ is compact in H^1(D_R). In fact, Let {ϕ_j} be a bounded sequence  in H^1(D_R) and the boundedness implies there exists a subsequence denoted also by  {ϕ_j} satisfying ϕ_j ⇀ϕ for some ϕ∈ H^1(D_R). On the other hand, ϕ_j →ϕ in L^2(D_R) due to the Rellich–Kondrachov theorem. From the definition of ℬ, we know that {Bϕ_j} is weakly convergent in H^1(D_R) and (ℬ(ϕ_j -ϕ),ψ) =a_2 ((ϕ_j -ϕ),ψ). Let ψ =ℬ(ϕ_j -ϕ),ψ), it is clear that

    ℬ(ϕ_j -ϕ)_H^1(D_R)⩽  4k^2 max{1+τ_L^∞(Ω),2}ϕ_j-ϕ_L^2(D_R)→ 0.

It means that ℬ is compact in H^1(D_R).

We claim that 𝒜 + B-(Λ-Λ_0) is bijective with a bounded inverse in H^1(D_R) if it is injective. In fact, since A^-1 is bijective and bounded, then

    𝒜 + B-(Λ-Λ_0) = A(I - (-A^-1(B-(Λ-Λ_0))),

implies the bijective equivalence of 𝒜 + B-(Λ-Λ_0) and I - (-A^-1(B-(Λ-Λ_0)). From the fact that A^-1 is bounded and B-(Λ-Λ_0) is compact, we can get that -A^-1(B-(Λ-Λ_0)) is compact. The Fredholm theory gives the bijection of  𝒜 + B-(Λ-Λ_0) with a bounded inverse if it is injective. Therefore, To show the existence and uniqueness it suffices to show that 𝒜 + B-(Λ-Λ_0)  is injective, i.e. the only solution of homogeneous integral equation (<ref>) or the equivalent homogeneous system (<ref>) is identically zero. If this is done, by the Lax-Milgram theory the integral equation (<ref>) can be inverted in H^1(D_R) and the inverse operator is bounded. From this, it follows that (u,u^s) depends on the boundary data (f,g).

The uniqueness of (<ref>) is guaranteed by the Rellich theorem. In specific, when f=g =0, we can get

    ∫_∂ D_R∂ u^s /∂νu^s   =( ∫_D_R \ΩΔ u^s u^s + |∇ u^s|^2 + ∫_∂Ω∂ u^s /∂νu^s)
       = ∫_∂Ωσ∂ u/∂νu = ∫_∂Ωσ|∇ u|^2 -  k^2∫_∂Ωτ|u|^2 ⩽ 0.

Then u^s  = 0 in ^N \Ω. The homogeneous boundary data now imply that u = ∂_ν u = 0 on ∂Ω. From the unique continuity principle <cit.> we get u = 0 in Ω. Summarizing all the above analysis, we have the continuous dependence on the boundary data of the incident field to the direct scattering problem in H^1(D_R).  The standard elliptic  regularity estimate can increase the regularity from H^1(D_R) to  H^2(D_R) if the boundary data (g,f) ∈ H^1/2(∂Ω) × H^3/2(∂Ω).
We observe that the Sobolev norm of u^i-v also gives the boundary information due to trace theorem.  Hence one has that
 
    u^i-v_H^3/2(∂Ω)⩽ε  and  ∂_ν u^i- ∂_ν v _H^1/2(∂Ω)⩽ε.

We let (w_j, w^s_j) be the unique solution to the system (<ref>) with the boundary conditions   f := u^i-v and g := ∂_ν u^i- ∂_ν v on ∂Ω and (g,f) ∈ H^1/2(∂Ω) × H^3/2(∂Ω).
Then it is clear that

    w_j^s_H^2_loc (^N \Ω)⩽  C(k,σ,τ)ε.

If the transmission eigenfunction v can be exactly  extended to v^i defined in the whole space, Then the scattered field w^s illuminated by v^i is identically zero due to the zero boundary conditions. By the linearity of (<ref>), we can further get  u^s = w_j^s + w^s and

    u^s_H^2_loc (^N \Ω)⩽w_j^s_H^2_loc (^N \Ω)  + w^s_H^2_loc (^N \Ω)⩽ Cε.






















































 §.§ Proof of Theorem <ref>


With all the preliminary work above, we are in a position to prove the main results. The main idea of this proof is based on the spectrala and geometric patterns of transmission eigenfunctions introduced in section <ref>.

We mainly consider the three-dimensional case, and the two-dimensional case can be proven similarly except for the transmission eigenfunction v_m^l substituted by v_m defined in (<ref>).

We first define an exterior domain around the  inhomogeneous medium inclusion:

    Γ_e(Ω,ϵ) = {x ∈^3 |  dist(x,Γ)⩽ϵ, x ∉Ω},

where Γ is any subset of ∂Ω and ϵ = o(1) is an any given distance parameter. We can artificially generate field concentration phenomenon in Γ_e(Ω,ϵ) by properly choosing an incident field u^i, namely ∇ u_L^∞(Γ_e(Ω,ϵ)) >M for any given large M. The construction of M shall be more definitely specified in what follows. In order to make the proof clearer, we divide it to several steps:

Step 1: We first construct the surface-localized transmission eigenfunctions. Let B_r_0∈^3 be a radial ball with radius r_0>0 satisfying

    ϵ : =dist(B_r_0,Γ) = o(1).

By choosing a  proper coordinate system, we assume  B_r_0 centers at the origin.
We consider the transmission eigenvalue problem (<ref>) in B_r_0.
From Lemma <ref>, there exists a sequence of material parameters (σ_m,τ_m)_m ∈  such that k is  the transmission eigenvalue.  For example, we can assume

    σ_m() = 1 τ_m() = √(n_m)∈ B_r_0,

where n_m is given in (<ref>).
The corresponding transmission eigenfunction v_m^l defined in (<ref>) is  surface-localized and satisfies the homogeneous Helmholtz equation (<ref>) in  B_r_0.

We further consider the transmission eigenvalue problem in (Ω; σ,τ) ∪ (B_r_0;σ_m,τ_m) where (σ,τ) and (σ_m,τ_m) are defined in (<ref>) and (<ref>) respectively. Lemma <ref> shows that the transmission eigenvalue k in B_r_0 is also a transmission eigenvalue in Ω∪ B_r_0.  Moreover, the corresponding transmission eigenfunction 𝐯_m^l is given by

    𝐯_m^l = v_m^l·χ__B_r_0 + 0·χ__Ω.

It is clear that 𝐯_m^l as the transmission eigenfunction  is the solution of homogeneous Helmholtz equation (<ref>) in Ω∪  B_r_0.

Step 2: We next construct the incident field with the Herglotz approximation.
In order to identify the transmission eigenfunction uniquely, we assume the normalized condition 𝐯_m^l_L^2(Ω∪ B_r_0) =1. Then

    β_m^l = 1/√(∫_0^r_0 j^2_m(kr)r^2r) = = √(2/π)k^3/2/√(∫_0^kr_0 J^2_m+1/2(r)rr).

Since Ω is a of class C^1,1 and Ω∪ B has a connected complement in ^N, hence lemma <ref> shows that there exists a Herglotz wave v_g,k such that v_g,k  is a H^2(Ω) approximation to 𝐯_m^l. In other words, given any ε >0, it holds that

    v_g,k - 𝐯_m^l_H^2(Ω∪ B_r_0)⩽ε.

The Herglotz function v_g,k is in fact the desired incident field since v_g,k is the entire solution of Helmholtz equation (<ref>) in the whole space. Moreover, the Herglotz function v_g,k exhibits strong resonant behaviors in  B_r_0 as the surface transmission eigenmode and nearly vanishes inside the inclusion Ω.
It is noted that the construction of v_g,k is based on the spectral and geometric patterns of transmission resonance. However, the existence of v_g,k does not depend on the existence of (B_r_0;σ_m,τ_m).  In fact, B_r_0 is an auxiliary radial domain with the same material parameter as the homogeneous background space. We denote the incident field by u^i := v_g,k.

Step 3: In the following, we analyze the behaviors of the total field u  illuminated by u^i in (<ref>). The inequality (<ref>) also indicates

    u^i - 𝐯_m^l_H^2(Ω)⩽ε.

From lemma <ref>, it is clear that the scattered field u^s illuminated by u^i would be sufficiently small outside Ω. With  the Sobolev embedding  H^2 ↪ C^0,   we can further get

    u^s_∞, G\Ω⩽ C(k,σ,τ) ε,

where G is an any bounded domain containing Ω∪ B_r_0. Therefore, the inclusion Ω is like invisible obstacle such that the behaviors of the incident field is almost undisturbed in the presence of the scatterer Ω.


Since the behaviors of total field  is almost described by the incident field outside the homogeneous inclusion Ω, we can investigate the behavior of incident field v_g,k. First,

    v_m^l() = β_m^l j_m(kr) Y_m^l(θ,ψ) =k/√(∫_0^kr_0 J^2_m+1/2(r)rr)J_m+1/2(kr)r^-1/2Y_m^l(θ,φ) ∈ B_r_0.

Without loss of generality, we always assume m is sufficiently large such that kr_0 is less than j'_m+1/2,1. Then J_m+1/2(kr)r^-1/2 is monotonously increasing for r ∈ (0,r_0) and attains the maximum ar r = r_0. By straight calculation, it holds that

    max_r∈ (0,r_0)J_m+1/2(kr)kr^-1/2/√(∫_0^kr_0 J^2_m+1/2(r)rr) = J_m+1/2(kr_0)kr_0^-1/2/√(∫_0^kr_0 J^2_m+1/2(r)rr) =  √(2m+3)/r_0^3/2(1 + 𝒪(m^-2) ).

and

    sup_(θ ,φ)∈ T  Y_m^l (θ,φ) = √(2m+1/4π)sup_θ∈ [0,π]√(%s/%s)(m-|l|)!(m-|l|)!P_m^|l|(cosθ) ⩾√(1.11/4πm+1/2/|l|+1)  -m ⩽ l ⩽ m,

where T := [0,π] × [φ_1,φ_2],φ_1,φ_2 ∈ [0,2π)  we have used lemma <ref>. Combining the above estimates, we can finally obtain

    sup_∈B_r_0 v_m^l() = sup_∈∂B_r_0 v_m^l()  ⩾1/√(2π)r_0^3/2m+1/2/√(|l|+1)(1 + 𝒪(m^-2) )⩾1/3r_0^3/2m+1/2/√(|l|+1),

for sufficiently large m.

Step 4: We finally can prove the gradient of total field would blow up inside Γ_e(Ω,ϵ). From the formulas (<ref>), (<ref>), (<ref>) and the Sobolev embedding  H^2 ↪ C^0 inside B_r_0∪Ω, it holds that

    sup_∈∂B_r_0 u^i()  ⩾1/3r_0^3/2m+1/2/√(|l|+1) - ε  and  sup_∈∂Ω u^i() ⩽ε.

The estimate (<ref>) implies that the scattered field  would be sufficiently small if we designate the incident field u^i to be the Herglotz approximation v_g,k to (<ref>). It means that the total field  outside the scatterer Ω equals to the incident field plus a sufficiently small term. Let

    _0 = sup_∈∂ B_r_0v_g,k().

we can assume ϵ  :=dist(B_r_0,Γ) = dist(_0,Γ) by rigid motions if necessary. Then

    u(_0) ⩾1/3r_0^3/2m+1/2/√(|l|+1) -(1+C(k,σ,τ)) ε  and  sup_∈∂Ω u() ⩽ (1+C(k,σ,τ))ε.

Since (1+C(k,σ,τ))ε is sufficiently small, then we can assume

    (1+C(k,σ,τ))ε⩽1/12r_0^3/2m+1/2/√(|l|+1).

Finally by using the mean value theorem, we can obtain the conclusion

    ∇ u _L^∞(Γ_e(Ω,ϵ))⩾1/ϵ1/6r_0^3/2m+1/2/√(|l|+1)⩾1/6r_0^-3/2 (m+1/2)^1/21/ϵ.

It is obvious from (<ref>) that ∇ u would blow up when m goes to infinity or ϵ goes to zero. Therefore,  For any given large number M, we can always construct the parameters ϵ,m and r_0 such that

    1/6r_0^-3/2 (m+1/2)^1/21/ϵ⩾ M.

Here, the construction of parameters ϵ,m and r_0 depends on the  incident field u^i.



It is noted that the field concentration can occur at multiple localizations near the inclusion Ω. In specific, we  can add multiple disjoint virtue field concentration generators B_1 ,⋯,B_L_0 around the inclusion Ω. The incident field u^i is designated to  Herglotz approximation to 𝐯_m^l = v_m^l·χ__B_1 + v_m^l·χ__B_2 + ⋯ + v_m^l·χ__B_L_0 +  0·χ__Ω where v_m^l are surface-localized transmission eigenfunctions up to a translation transformation. Thus,  the field concentration can occur in the intervals between Ω and B_1 ,⋯,B_N by following a similar argument.






From the theorem <ref>, it is obvious that we can achieve field concentration in two customized ways. One is high-order probing. We can choose a high-order field concentration generator with  more intensive surface-localized property such that m is sufficiently large. The other is in a geometric manner, i.e. the resonator is sufficiently close to the inclusion (ϵ→ 0).










§ NUMERICAL EXPERIMENTS

In this section,  several numerical experiments  are presented to verify the theoretical results.  On the one hand, we shall determine the Herglotz wave, which is an approximation to transmission eigenfunction localizing on the surface of B while almost vanishing within the interior of Ω. On the other hand, we utilize the Herglotz wave as the incident field and obtain the total field by solving the scattering problem of the Helmholtz equation.


Given   a radial area B near the scatterer Ω,   we first obtain kernel function g(x) by  solving the integral equation (<ref>). Noting that H_k is compact, that is, the inverse of integral equation (<ref>) is generally ill-posed,  then we use the Tikhonov regularization method as introduced in <cit.>. We consider the integral equation
H_k g = f    f  ∈ L^2 (∂Ω) × L^2(∂ B). The Tikhonov regularization is adopted to the following equation:

    α g_α +H^*_gH_g g_α = H^*_g f.

Thus, the regularized kernel function g_α is given by (α + H^*_gH_g)^-1H^*_g f. Next, we extend the  incident field to the whole domain by using the Herglotz approximation, namely, u^i=H_kg_α.
Finally, we designate u^i as the incident field and solve the inhomogeneous scattering system (<ref>).

In what follows, we present three examples to illustrate the phenomenon of field concentration.
In the first example, we consider an ellipse domain Ω, i.e.,

    Ω:={z=(x,y)  | (x+3.2)^2/3^2 + (y+3.2)^2/4^2 < 1}.

 Here, we set the wavenumber k=3 and the parameter  σ=1.  By solving the transmission eigenvalue problem (<ref>),  we obtain  τ=17.5285, where the regularization parameter is chosen as α=10^-5.
 The second example is an rectangle with width a=4 and height b=8, that is,

    Ω:={z=(x,y) |  -5.1 <  x < 1.1 , -4 < y < 4 },

where the parameters are set to be k=1 and σ=1/4. Furthermore,  we  have τ=16.2183,  where the regularization parameter is chosen as α=10^-6.
In the final example, we consider a kite domain Ω, i.e.,

    Ω:={z=(x,y)  |  (x,y)=(cos t+0.65cos 2t+1.4,  1.5 sin t),  t∈[0,  2π]}.

 Here, the parameters are set to be k=√(3) and σ=1/3,  thus we get τ=19.9121  and the regularization parameter is chosen as α=10^-3.


		


 From figures  <ref>(a), <ref>(a) and <ref>(a),  one can find that the total field  is bounded and is a sufficiently precise approximation to the surface-localized eigenfunction and trivial eigenfunction in the area B and Ω respectively. Furthermore,  it can be observed that the gradient field ∇ u blows up near the scatterer and would decay sharply when the total field leaves the area B, as depicted in figures <ref>(b), <ref>(b) and <ref>(b). To exhibit the imaging results clearly, we enlarge a portion of the image (the red dashed square domain in figures <ref>(b), <ref>(b) and <ref>(b)). It is clear to see that the gradient of the total field demonstrates the field concentration phenomenon in figures <ref>(c),  <ref>(c) and <ref>(c). It is worth mentioning that the dashed circles presented in the figures are virtual objects and do not exist in practical applications. Moreover, there are no restrictions on the scale of the circles, and they can be of arbitrary scale. Therefore, our method can demonstrate good performance for multi-scaled scatterers.





§ SURFACE-OSCILLATING OF TRANSMISSION EIGENFUNCTIONS

In general,  the magnitude of  a wave field gradient  is proportional to its wavenumber,  as observed in plane waves  such as the plane wave e^ k · d where d ∈𝕊^N-1. However, this relationship does not hold for surface-localized transmission resonant eigenfunctions. The oscillating behaviors of these transmission eigenfunctions tends to concentrate in the neighborhood along the boundary and the oscillating frequencies are much higher than the excitation frequency.




 §.§ Auxiliary results

Before proving Theorem <ref>, we first introduce several auxiliary lemmas to facilitate the main proof.

The following integral formula for any m>0 and y ⩾ 0 holds:

    ∫_0^y J'^2_m(x)x + m^2/xJ_m^2(x) x = ∫_0^y J^2_m-1(x)xx - m J_m^2(y).



The recurrence J'_m(x) = J_m-1(x)-m/xJ_m(x) gives

    ∫_0^y J'^2_m(x)x + m^2/xJ_m^2(x) x   = ∫_0^y ( J_m-1(x)- m/xJ_m(x))^2x + m^2/xJ_m^2(x) x
       = ∫_0^y J^2_m-1(x)x -2mJ_m(x)J'_m(x)x .

Then using the integration by part again we get the equality.


<cit.>
For real x ∈ [-1,1] and integer l,m with 1 ⩽ l ⩽ m, it holds that

    1/√(2.22(l+1))  < max_x ∈ [-1,1]| P(m,l;x) |:=max_x ∈ [-1,1]|√((m-l)!/(m+l)!)P_m^l(x)|  < 2^5/4/π^3/41/l^1/4.

There exists an absolute maximum x_0∈ [0,1) such that |P_m^l(x_0)| =  max_x∈ [-1,1]|P_m^l(x)| satisfying

    1 - (1.11(l+1))^2/(m+1/2)^2⩽ x_0 ⩽ 1- l^2/m(m+1).






 §.§ proof of Theorem <ref>


The existence of a sequence of transmission eigenfunctions and their surface-localization have been proved in Lemma <ref>. We focus on the surface oscillating behaviors of transmission eigenfunctions  here. The transmission eigenfunctions share the same mathematical representations for any given radius r_0 and the proof is similar. For simplicity, we assume r_0 = 1.

The transmission eigenfunctions (v_m,w_m) in ^2 are given in the formula (<ref>). The gradient formula in polar coordinates yields

    ∇ v_m    =  β_m (kJ'_m(kr)r̂ +im/rJ_m(kr)θ̂)e^imθ , 
    ∇ w_m    =  α_m (kn_mJ'_m(kn_mr)r̂ +im/rJ_m(kn_mr)θ̂)e^imθ ,

Where (r̂,θ̂) is a pair of unit orthonormal vector in polar coordinates.
By using the orthogonality of {e^ m θ}_m ∈ in the unit circle, one has

    ∇ v_m_L^2(B_ξ)^2 =β_m^2∫_0^kξ  J'^2_m(r)r +m^2/rJ^2_m(r) r.

By using the asymptotic formulas of Bessel functions (<ref>) when m is sufficiently large, it holds that

    ∇ v_m_L^2(B_ξ)^2/∇ v_m_L^2(B_1)^2 = ∫_0^kξ  J'^2_m(r)r +m^2/rJ^2_m(r) r/∫_0^k  J'^2_m(r)r +m^2/rJ^2_m(r) r∼∫_0^kξ r^2m-1r/∫_0^k r^2m-1r∼ξ^2m→ 0  m →∞.

So {∇ v_m} are also surface-localized especially for sufficiently m. For w_m,

    ∇ w_m_L^2(B_ξ)^2    = α_m^2 ∫_0^ξ( k^2n_m^2J'^2_m(kn_mr) + m^2/r^2J^2_m(kn_mr))r r
       = α_m^2 ∫_0^kn_mξ  J^2_m-1(r)r r - mJ_m^2(kn_mξ).

where we have used the Lemma <ref>.
By following a similar argument as (<ref>), it holds that

    ∫_0^kn_mξ  J^2_m-1(r)r r - mJ_m^2(kn_mξ) ⩽1/2J^2_m-1(kn_mξ) (kn_mξ)^2,

and

    ∫_0^kn_m  J^2_m-1(r)r r   - mJ_m^2(kn_mξ)  ⩾∫_0^m-1  J^2_m-1(r)r r - mJ_m^2(m-1) 
       ⩾1/2(m-1)^2J_m-1^3(m-1)/J_m-1(m-1)+2(m-1)J'_m-1(m-1)-  mJ_m-1^2(m-1),

where we have used the fact J_m(m-1) <J_m-1(m-1). Hence,

    ∇ w_m_L^2(B_ξ)^2/∇ w_m_L^2(B_1)^2   =  ∫_0^kn_mξ  J^2_m-1(r)r r - mJ_m^2(kn_mξ)/∫_0^kn_m  J^2_m-1(r)r r - mJ_m^2(kn_m)
       ⩽J^2_m-1(kn_mξ) (kn_mξ)^2/(m-1)^2J_m-1^3(m-1)/J_m-1(m-1)+2(m-1)J'_m-1(m-1)-  2mJ_m-1^2(m-1)
       ⩽J^2_m-1(kn_mξ)/J_m-1^2(m-1) (kn_mξ)^2/(m-1)^21/J_m-1(m-1)/J_m-1(m-1)+2(m-1)J'_m-1(m-1)-2m/(m-1)^2.


From the asymptotic formula for J_m-1(m-1) and J'_m-1(m-1) (see 9.3.31 in <cit.>), we can obtain

    J_m-1(m-1)/J_m-1(m-1)+2(m-1)J'_m-1(m-1)-2m/(m-1)^2
    = (m-1)^-2/3(C + (m-1)^-1/3) ,

where C is some positive constant.
Note that kn_mξ <(m-1) and following a similar argument as (<ref>), we can readily get

    ∇ w_m_L^2(B_ξ)/∇ w_m_L^2(B_1)   ⩽ C (m-1)^2/3Ai((m-1)^2/3ζ_ξ) ⩽ C (m-1)^1/2 e^-2/3ζ_ξ^3/2(m-1)→ 0  m →∞.

where C is some constant  only depending on k,ξ. Then {∇ w_m}_m ∈ are also surface-localized.



We now prove the oscillating frequencies of (v_m,w_m) are much higher than the excitation frequency k.
The normalized condition v_m_L^2(B_1) = 1 shows that

    ∇ v_m =   β_m (kJ'_m(kr)r̂ +im/rJ_m(kr)θ̂)e^imθ
    
    =   1/√(2π)kJ_m(k)/√(∫_0^kJ_m^2(r)rr)(kJ'_m(kr)/J_m(k)r̂ +im/rJ_m(kr)/J_m(k)θ̂)e^imθ

First, the formula of (<ref>) yields

    kJ_m(k)/√(∫_0^kJ_m^2(r)rr)∼k^m+1/√(1/2(m+1)k^2m+2)∼ 2√(m+1).

Moreover, The Bessel function J_m(x) and its derivative  J'_m(x) are strictly monotonously increasing. Then ∇ v_m attains the maximum at r=1. By using the asymptotic formula (<ref>) again, it readily holds that

    ∇ v_m_L^∞(Σ_ξ)/k⩾ C_k,s_0 m^3/2→∞  as    m →∞.

It is remarked that the lower bound (<ref>) is in fact optimal and ∇ v_m attains the supremum on the boundary r=1.

For ∇ w_m, the relation between α_m and β_m together with the  normalized condition v_m_L^2(B_1) = 1 gives

    ∇ w_m    =  α_m (kn_mJ'_m(kn_mr)r̂ +im/rJ_m(kn_mr)θ̂)e^imθ
       = 1/√(2π)kJ_m(k)/√(∫_0^kJ_m^2(r)rr)(kn_mJ'_m(kn_mr)/J_m(kn_m)r̂ +im/rJ_m(kn_mr)/J_m(kn_m)θ̂)e^imθ.

The maximum formula of Bessel functions and their derivatives

    max_x∈ J_m(x) = J_m(j'_m,1) ∼ m^-1/3max_x∈J'_m(x)∼ m^-2/3,

together with (<ref>), (<ref>) yields

    ∇ v_m_L^∞(Σ_ξ)/k   = k/√(2π)e^imθ/√(∫_0^kJ_m^2(r)rr)J_m(k)/J_m(kn_m)n_mJ_m'(kn_mr)r̂+ im/rkJ_m(kn_mr)θ̂_L^∞(Σ_ξ)
       ⩾ C_k,s_0 m^3/2→∞  as   m →∞.

It is remarked that  ∇ w_m attains the maximum at  r_m :=j'_m,1/kn_m where r_m → 1 as m→∞. This is a circle with the radius r_m which is very close but not touch the boundary r=1.


The entire proof in three dimensions follows a similar structure to that in two dimensions, with some necessary modifications due to technical calculations. We will outline the required modifications in the subsequent sections.


The corresponding gradient ∇ v_m in spherical coordinate is

    ∇ v_m^l = ∂ v_m^l/∂ rr̂ + ∇_(θ,ψ) v_m^l = β_m^l( k j'_m(kr)Y_m^lr̂ +  j_m+1/2(kr) ∇_(θ,ψ)Y_m^l).

Since r(m(m+1))^-1/2∇_(θ,ψ)Y_m^l are a sequence of  orthonormal  vector spherical harmonics, then

    ∇ v_m^l ^2_L^2(B_ξ) =    (β_m^l)^2 (∫_0^ξ j'^2_m(kr)k^2r^2 + m(m+1)j^2_m(kr) r) 
       =(β_m^l)^2/k(∫_0^kξ j'^2_m(r)r^2 + m(m+1)j^2_m(r) r) .

Then

    ∇ v_m^l ^2_L^2(B_ξ)/∇ v_m^l ^2_L^2(B_1)   = ∫_0^kξ j'^2_m(r)r^2 + m(m+1)j^2_m(r) r/∫_0^k j'^2_m(r)r^2 + m(m+1)j^2_m(r) r
       ∼∫_0^kξr^2mr/∫_0^kr^2mr∼ξ^2m+1→ 0   as   m →∞.

Similarly, the representation of ∇ w_m^l is

    ∇ w_m^l = ∂ w_m^l/∂ rr̂ + ∇_(θ,ψ) w_m^l = α_m^l( kn_m j'_m(kn_mr)Y_m^lr̂ +  j_m(kn_mr) ∇_(θ,ψ)Y_m^l).

With the relation j_m(x) = √(π/2x)J_m+1/2(x), the L^2-norm of ∇ w_m^l is given by

    ∇ w_m^l ^2_L^2(B_ξ) =    (α_m^l)^2 (∫_0^ξ j'^2_m(kn_mr)k^2n_m^2r^2 + m(m+1)j^2_m(kr) r) 
       = π/2(α_m^l)^2/kn_m(∫_0^kn_mξ J'^2_m+1/2(r)r + (m+1/2)^2/rJ^2_m+1/2(r) r - 1/2J^2_m+1/2(kn_mξ)) 
       =π/2(α_m^l)^2/kn_m( ∫_0^kn_mξ J^2_m-1/2(r)rr - (m+1)J^2_m+1/2(kn_mξ)  ).

By using the result in (<ref>), it holds that

    ∇ w_m^l ^2_L^2(B_ξ)/∇ w_m^l ^2_L^2(B_1)   =∫_0^kn_mξ J^2_m-1/2(r)rr - (m+1)J^2_m+1/2(kn_mξ)/∫_0^kn_m J^2_m-1/2(r)rr - (m+1)J^2_m+1/2(kn_mξ)
       ⩽∫_0^kn_mξ J^2_m-1/2(r)rr - (m+1)J^2_m+1/2(kn_mξ)/∫_0^m -1/2 J^2_m-1/2(r)rr - (m+1)J^2_m-1/2(m-1/2)
       =  J^2_m-1/2(kn_mξ)/J_m-1/2^2(m-1/2)(kn_mξ)^2/(m-1/2)^21/J_m-1/2(m-1/2)/J_m-1/2(m-1/2)+2(m-1/2) J'_m-1/2(m-1/2) - 2 m+1/(m-1/2)^2
       ⟶ 0   as   m →∞.



In order to prove the oscillating frequencies of (v_m^l,w_m^l) are much higher than the excitation frequency k, we need a clearer representation for the transmission eigenfunction. First,  The normalized spherical harmonics Y_m^l has the explicit expression ,

    Y_m^l(θ,φ) = √(2m+1/4π(m-|l|)!/(m+|l|)!)P_m^|l|(cosθ) e^ilφθ∈ [0,π] , φ∈ [0,2π) , -m⩽ l ⩽ m,

where P_m^|l| are the associated Legendre polynomials.  The normalized condition v_m^l_L^2(B_1)=1 gives

    β_m^l =  1/√(∫_0^1 j^2_m(kr)r^2 r)  = 2/πk/√(∫_0^k J^2_m+1/2(r)r r).

Moreover,  the corresponding gradient ∇ v_m^l in spherical coordinates is


    ∇ v_m^l =   ∂ v_m^l/∂ rr̂ +1/r∂ v_m^l/∂θθ̂ + 1/r sinθ∂ v_m^l/∂φφ̂
    
    =   β_m^l√(2m+1/4π(m-|l|)!/(m+|l|)!)r^-3/2e^ilφ[ (-1/2J_m+1/2(kr)+krJ'_m+1/2(kr) ) P_m^|l|(cosθ)r̂.  
       .  + J_m+1/2(kr)P_m^|l|(cosθ)/θθ̂ + J_m+1/2(kr)P_m^|l|(cosθ) l/sinθφ̂]  
    
    :=   β_m^lJ_m+1/2(k)√(2m+1/4π)(I_1r̂ +I_2θ̂ + I_3φ̂),


where r̂ , θ̂ ,  φ̂ are the orthonormal unit vectors in spherical coordinates.























Next, we need to estimate I_i,i=1,2,3 in the above formula. By using Lemma <ref>, we first have

    sup_x ∈Σ_ξI_1   = sup_∈Σ_ξ√((m-|l|)!/(m+|l|)!)1/J_m+1/2(k)| r^-3/2e^ilφ(krJ'_m+1/2(kr)-1/2J_m+1/2(kr))P_m^|l|(cosθ) |
       ⩾  Cm+1/2/(|l|+1)^1/2 |l| ⩾ 1,

where we used the asymptotic formula of Bessel functions and its derivatives for large order.
It is clear that (<ref>) also holds for l=0 since the Legendre polynomials max_x ∈ [-1,1] P_m(x) =1. For I_3, we shall calculate the value at maximum point x_0 to get the lower bound. Let x_0 =cosθ_0, it holds from Lemma <ref> that

    sinθ_0 = √(1-x_0^2)⩽1.11(|l|+1)/m+1/2 1 ⩽ |l|⩽ m.

Then

    sup_∈Σ_ξ I_3    ⩾sup_x ∈Σ_ξ√((m-|l|)!/(m+|l|)!)1/J_m+1/2(k)| r^-3/2e^ilφJ_m+1/2(kr)(sinθ_0)^-1lP_m^|l|(cosθ_0)| 
       ⩾ Cm+1/2/(|l|+1)^1/2|l|/|l|+1 |l| ⩾ 1,

and sup_∈Σ_ξ I_3 =0 when l=0.

We shall estimate I_2 in several cases. When 2 ⩽ |l| ⩽ m-1, we estimate the upper bound of the derivative of the associated Legendre polynomial. It holds that

    sup_θ∈ [0,π]   |√((m-|l|)!/(m+|l|)!)P_m^|l|(cosθ)/θ| =  sup_θ∈ [0,π]|√((m-|l|)!/(m+|l|)!)P_m^|l|(cosθ)/(cosθ)sinθ| 
       =  sup_θ∈ [0,π]1/2√((m-|l|)!/(m+|l|)!)|(m+|l|)(m-|l|+1)P_m^|l|-1(cosθ) - P_m^|l|+1(cosθ) | 
       =sup_θ∈ [0,π]√((m-|l|)!/(m+|l|)!)||l|cosθ/sinθP_m^|l|(cosθ) -  P_m^|l|+1(cosθ) | 
       ⩽sup_θ∈ [0,π]√((m-|l|)!/(m+|l|)!)||l|/sinθP_m^|l|(cosθ) | +C (m+1/2) sup_θ∈ [0,π]| P(m,|l|;cosθ)|.

where we have used the fact  that the maximum of P(m,|l|+1;cosθ) is uniformly less than that of P(m,|l|;cosθ) with respect to m and l in the last line <cit.>.
For other cases |l|=m,1   0, straight calculations show that

    sup_θ∈ [0,π]|√(1/(2m)!)P^m_m(cosθ)/θ|    = √(2m)sup_θ∈ [0,π]| P(m,m-1;cosθ) | 
    sup_θ∈ [0,π]√((m-1)!/(m+1)!)|P^1_m(cosθ)/θ|    = sup_θ∈ [0,π]|P_m(cosθ) - √((m+2)(m-1))P(m,2;cosθ)|.

Comparing the formulas (<ref>),(<ref>), (<ref>) with the formula (<ref>), one can find that the upper bound of I_2 can be uniformly controlled by that of I_3 when |l| ⩾ 1, i.e.

    sup_x ∈Σ_ξ I_2 =sup_∈Σ_ξ√((m-|l|)!/(m+|l|)!)1/J_m+1/2(k)| r^-3/2e^ilφJ_m+1/2(kr)P_m^|l|(cosθ)/θ| ⩽ C sup_∈Σ_ξ I_3  .

While l=0, it holds that

    sup_θ∈ [0,π]|P_m(cosθ)/θ| = √(m(m+1))sup_θ∈ [0,π]|P(m,1;cosθ) |.

Combining (<ref>) , (<ref>), (<ref>), (<ref>) (<ref>), one can deduce that

    ∇ v_m^l_L^∞(Σ_ξ)/k   =J_m+1/2(k)/√(∫_0^1 J_m+1/2(kr)r r)√(2m+1/4π k^2)√((I_1)^2 +(I_2)^2 + (I_3)^2)_L^∞(Σ_2)
       ⩾ C(m+1/2)^2/(|l|+1)^1/2⩾ C(m+1/2)^3/2→∞.

when m →∞. In fact, ∇ v_m^l attains the supremum on the boundary sphere r =1.


From the relation between α_m^l and β_m^l, we can obtain

    α_m^l = j_m(k)/j_m(kn_m)β_m^l = √(n_m)2/πJ_m+1/2(k)/J_m+1/2(kn_m)k/√(∫_0^k J^2_m+1/2(r)r r).


We can get the coefficient of w_m^l through the relation of α_m^l and β_m^l. Then the gradient of w_m^l in spherical coordinates is

    ∇ w_m^l =   α_m^l √(2m+1/4π(m-|l|)!/(m+|l|)!)r^-3/2e^ilφ[ ( kn_mrJ'_m+1/2(kn_mr) -1/2J_m+1/2(kn_mr) )P_m^|l|(cosθ)r̂
         + J_m+1/2(kn_mr)P_m^|l|(cosθ)/θθ̂ +  J_m+1/2(kn_mr)P_m^|l|(cosθ) l/sinθφ̂] 
       :=α_m^l√(2m+1/4π)(L_1r̂ +L_2θ̂ + L_3φ̂).

 We note the fact

    kn_m ∈ (j_m+1/2,s_0,j_m+1/2,s_0+1) = m+1/2 + a_s_0(m+1/2)^1/3 + 𝒪((m+1/2)^-1/3) >j_m+1/2,1 > j'_m+1/2,1,

where a_s_0 is some positive constant. Then J_m+1/2(x)

    max_x∈ J_m+1/2(x) = J_m+1/2(j'_m+1/2,1) ∼ (m+1/2)^-1/3  max_x∈J'_m+1/2(x)∼ (m+1/2)^-2/3.

 Hence,  we can conduct similar estimates as I_i for L_i,i=1,2,3 to get the lower gradient bound and

    ∇ w_m^l_L^∞(Σ_ξ)/k⩾ C(m+1/2)^13/6/(|l|+1)^1/2⩾ C (m +1/2)^5/3→∞  as   m →∞.

where C is a constant independent of m and l.
















§ ACKNOWLEDGMENT

We are grateful to  Huaian Diao and Yan Jiang  for their insightful
comments during discussions about this work.

abbrv














