Trace operator

In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.
Motivation.
On a bounded, smooth domain formula_1, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:
with given functions formula_3 and formula_4 with regularity discussed in the application section below. The weak solution formula_5 of this equation must satisfy
The formula_8-regularity of formula_9 is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense formula_9 can satisfy the boundary condition formula_11 on formula_12: by definition, formula_13 is an equivalence class of functions which can have arbitrary values on formula_12 since this is a null set with respect to the n-dimensional Lebesgue measure.
If formula_15 there holds formula_16 by Sobolev's embedding theorem, such that formula_9 can satisfy the boundary condition in the classical sense, i.e. the restriction of formula_9 to formula_12 agrees with the function formula_4 (more precisely: there exists a representative of formula_9 in formula_22 with this property). For formula_1 with formula_24 such an embedding does not exist and the trace operator formula_25 presented here must be used to give meaning to formula_26. Then formula_5 with formula_28 is called a weak solution to the boundary value problem if the integral equation above is satisfied. For the definition of the trace operator to be reasonable, there must hold formula_29 for sufficiently regular formula_9.
Trace theorem.
The trace operator can be defined for functions in the Sobolev spaces formula_31 with formula_32, see the section below for possible extensions of the trace to other spaces. Let formula_1 for formula_34 be a bounded domain with Lipschitz boundary. Then there exists a bounded linear trace operator
such that formula_25 extends the classical trace, i.e.
The continuity of formula_25 implies that 
with constant only depending on formula_42 and formula_43. The function formula_44 is called trace of formula_9 and is often simply denoted by formula_26. Other common symbols for formula_25 include formula_48 and formula_49.
Construction.
This paragraph follows Evans, where more details can be found, and assumes that formula_43 has a formula_51-boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo. On a formula_51-domain, the trace operator can be defined as continuous linear extension of the operator
to the space formula_54. By density of formula_55 in formula_54 such an extension is possible if formula_25 is continuous with respect to the formula_54-norm. The proof of this, i.e. that there exists formula_59 (depending on formula_43 and formula_42) such that
is the central ingredient in the construction of the trace operator. A local variant of this estimate for formula_64-functions is first proven for a locally flat boundary using the divergence theorem. By transformation, a general formula_51-boundary can be locally straightened to reduce to this case, where the formula_51-regularity of the transformation requires that the local estimate holds for formula_64-functions.
With this continuity of the trace operator in formula_55 an extension to formula_54 exists by abstract arguments and formula_70 for formula_41 can be characterized as follows. Let formula_72 be a sequence approximating formula_41 by density. By the proven continuity of formula_25 in formula_55 the sequence formula_76 is a Cauchy sequence in formula_77 and formula_78 with limit taken in formula_77.
The extension property formula_29 holds for formula_81 by construction, but for any formula_38 there exists a sequence formula_72 which converges uniformly on formula_84 to formula_9, verifying the extension property on the larger set formula_86.
The case p = ∞.
If formula_43 is bounded and has a formula_51-boundary then by Morrey's inequality there exists a continuous embedding formula_89, where formula_90 denotes the space of Lipschitz continuous functions. In particular, any function formula_91 has a classical trace formula_92 and there holds
Functions with trace zero.
The Sobolev spaces formula_94 for formula_32 are defined as the closure of the set of compactly supported test functions formula_96 with respect to the formula_54-norm. The following alternative characterization holds:
where formula_99 is the kernel of formula_25, i.e. formula_101 is the subspace of functions in formula_54 with trace zero.
Image of the trace operator.
For p &gt; 1.
The trace operator is not surjective onto formula_77 if formula_104, i.e. not every function in formula_77 is the trace of a function in formula_54. As elaborated below the image consists of functions which satisfy a formula_107-version of Hölder continuity.
Abstract characterization.
An abstract characterization of the image of formula_25 can be derived as follows. By the isomorphism theorems there holds
where formula_110 denotes the quotient space of the Banach space formula_111 by the subspace formula_112 and the last identity follows from the characterization of formula_101 from above. Equipping the quotient space with the quotient norm defined by
the trace operator formula_25 is then a surjective, bounded linear operator
Characterization using Sobolev–Slobodeckij spaces.
A more concrete representation of the image of formula_25 can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the formula_107-setting. Since formula_12 is a "(n-1)"-dimensional Lipschitz manifold embedded into formula_120 an explicit characterization of these spaces is technically involved. For simplicity consider first a planar domain formula_121. For formula_122 define the (possibly infinite) norm
which generalizes the Hölder condition formula_124. Then
equipped with the previous norm is a Banach space (a general definition of formula_126 for non-integer formula_127 can be found in the article for Sobolev-Slobodeckij spaces). For the "(n-1)"-dimensional Lipschitz manifold formula_12 define formula_129 by locally straightening formula_12 and proceeding as in the definition of formula_131.
The space formula_129 can then be identified as the image of the trace operator and there holds that
is a surjective, bounded linear operator.
For p = 1.
For formula_134 the image of the trace operator is formula_135 and there holds that
is a surjective, bounded linear operator.
Right-inverse: trace extension operator.
The trace operator is not injective since multiple functions in formula_54 can have the same trace (or equivalently, formula_138). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for formula_139 there exists a bounded, linear trace extension operator
using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that
and, by continuity, there exists formula_59 with
Notable is not the mere existence but the linearity and continuity of the right inverse. This trace extension operator must not be confused with the whole-space extension operators formula_145 which play a fundamental role in the theory of Sobolev spaces.
Extension to other spaces.
Higher derivatives.
Many of the previous results can be extended to formula_146 with higher differentiability formula_147 if the domain is sufficiently regular. Let formula_148 denote the exterior unit normal field on formula_12. 
Since formula_26 can encode differentiability properties in tangential direction only the normal derivative formula_151 is of additional interest for the trace theory for formula_152. Similar arguments apply to higher-order derivatives for formula_153.
Let formula_139 and formula_1 be a bounded domain with formula_156-boundary. Then there exists a surjective, bounded linear higher-order trace operator
with Sobolev-Slobodeckij spaces formula_158 for non-integer formula_127 defined on formula_12 through transformation to the planar case formula_161 for formula_121, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator formula_163 extends the classical normal traces in the sense that
Furthermore, there exists a bounded, linear right-inverse of formula_163, a higher-order trace extension operator
Finally, the spaces formula_168, the completion of formula_96 in the formula_146-norm, can be characterized as the kernel of formula_163, i.e.
Less regular spaces.
No trace in "Lp".
There is no sensible extension of the concept of traces to formula_173 for formula_32 since any bounded linear operator which extends the classical trace must be zero on the space of test functions formula_96, which is a dense subset of formula_173, implying that such an operator would be zero everywhere.
Generalized normal trace.
Let formula_177 denote the distributional divergence of a vector field formula_178. For formula_139 and bounded Lipschitz domain formula_1 define
which is a Banach space with norm
Let formula_148 denote the exterior unit normal field on formula_12. Then there exists a bounded linear operator
where formula_186 is the conjugate exponent to formula_42 and formula_188 denotes the continuous dual space to a Banach space formula_111, such that formula_190 extends the normal trace formula_191 for formula_192 in the sense that
The value of the normal trace operator formula_194 for formula_195 is defined by application of the divergence theorem to the vector field formula_196 where formula_197 is the trace extension operator from above.
"Application." Any weak solution formula_5 to formula_199 in a bounded Lipschitz domain formula_1 has a normal derivative in the sense of formula_201. This follows as formula_202 since formula_203 and formula_204. This result is notable since in Lipschitz domains in general formula_205, such that formula_206 may not lie in the domain of the trace operator formula_25.
Application.
The theorems presented above allow a closer investigation of the boundary value problem
on a Lipschitz domain formula_1 from the motivation. Since only the Hilbert space case formula_210 is investigated here, the notation formula_8 is used to denote formula_212 etc. As stated in the motivation, a weak solution formula_5 to this equation must satisfy formula_28 and
where the right-hand side must be interpreted for formula_217 as a duality product with the value formula_218.
Existence and uniqueness of weak solutions.
The characterization of the range of formula_25 implies that for formula_28 to hold the regularity formula_221 is necessary. This regularity is also sufficient for the existence of a weak solution, which can be seen as follows. By the trace extension theorem there exists formula_222 such that formula_223. Defining formula_224 by formula_225 we have that formula_226 and thus formula_227 by the characterization of formula_228 as space of trace zero. The function formula_227 then satisfies the integral equation
Thus the problem with inhomogeneous boundary values for formula_9 could be reduced to a problem with homogeneous boundary values for formula_224, a technique which can be applied to any linear differential equation. By the Riesz representation theorem there exists a unique solution formula_224 to this problem. By uniqueness of the decomposition formula_235, this is equivalent to the existence of a unique weak solution formula_9 to the inhomogeneous boundary value problem.
Continuous dependence on the data.
It remains to investigate the dependence of formula_9 on formula_3 and formula_4. Let formula_240 denote constants independent of formula_3 and formula_4. By continuous dependence of formula_224 on the right-hand side of its integral equation, there holds
and thus, using that formula_245 and formula_246 by continuity of the trace extension operator, it follows that
and the solution map
is therefore continuous.