On a bounded, smooth domain ${\textstyle \Omega \subset \mathbb {R} ^{n}}$ , consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:

${\displaystyle {\begin{alignedat}{2}-\Delta u&=f&\quad &{\text{in }}\Omega ,\\u&=g&&{\text{on }}\partial \Omega \end{alignedat}}}$ 

with given functions ${\textstyle f}$  and ${\textstyle g}$  with regularity discussed in the application section below. The weak solution ${\textstyle u\in H^{1}(\Omega )}$  of this equation must satisfy

${\displaystyle \int _{\Omega }\nabla u\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x}$  for all ${\textstyle \varphi \in H_{0}^{1}(\Omega )}$ .

The ${\textstyle H^{1}(\Omega )}$ -regularity of ${\textstyle u}$  is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense ${\textstyle u}$  can satisfy the boundary condition ${\textstyle u=g}$  on ${\textstyle \partial \Omega }$ : by definition, ${\textstyle u\in H^{1}(\Omega )\subset L^{2}(\Omega )}$  is an equivalence class of functions which can have arbitrary values on ${\textstyle \partial \Omega }$  since this is a null set with respect to the n-dimensional Lebesgue measure.

If ${\textstyle \Omega \subset \mathbb {R} ^{1}}$  there holds ${\textstyle H^{1}(\Omega )\hookrightarrow C^{0}({\bar {\Omega }})}$  by Sobolev's embedding theorem, such that ${\textstyle u}$  can satisfy the boundary condition in the classical sense, i.e. the restriction of ${\textstyle u}$  to ${\textstyle \partial \Omega }$  agrees with the function ${\textstyle g}$  (more precisely: there exists a representative of ${\textstyle u}$  in ${\textstyle C({\bar {\Omega }})}$  with this property). For ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  with ${\textstyle n>1}$  such an embedding does not exist and the trace operator ${\textstyle T}$  presented here must be used to give meaning to ${\textstyle u|_{\partial \Omega }}$ . Then ${\textstyle u\in H^{1}(\Omega )}$  with ${\textstyle Tu=g}$  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 ${\textstyle Tu=u|_{\partial \Omega }}$  for sufficiently regular ${\textstyle u}$ .

The trace operator can be defined for functions in the Sobolev spaces ${\textstyle W^{1,p}(\Omega )}$  with ${\textstyle 1\leq p<\infty }$ , see the section below for possible extensions of the trace to other spaces. Let ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  for ${\textstyle n\in \mathbb {N} }$  be a bounded domain with Lipschitz boundary. Then^[1] there exists a bounded linear trace operator

${\displaystyle T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega )}$ 

such that ${\textstyle T}$  extends the classical trace, i.e.

${\displaystyle Tu=u|_{\partial \Omega }}$  for all ${\textstyle u\in W^{1,p}(\Omega )\cap C({\bar {\Omega }})}$ .

The continuity of ${\textstyle T}$  implies that

${\displaystyle \|Tu\|_{L^{p}(\partial \Omega )}\leq C\|u\|_{W^{1,p}(\Omega )}}$  for all ${\textstyle u\in W^{1,p}(\Omega )}$ 

with constant only depending on ${\textstyle p}$  and ${\textstyle \Omega }$ . The function ${\textstyle Tu}$  is called trace of ${\textstyle u}$  and is often simply denoted by ${\textstyle u|_{\partial \Omega }}$ . Other common symbols for ${\textstyle T}$  include ${\textstyle tr}$  and ${\textstyle \gamma }$ .

Construction edit

This paragraph follows Evans,^[2] where more details can be found, and assumes that ${\textstyle \Omega }$  has a ${\textstyle C^{1}}$ -boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.^[1] On a ${\textstyle C^{1}}$ -domain, the trace operator can be defined as continuous linear extension of the operator

${\displaystyle T:C^{\infty }({\bar {\Omega }})\to L^{p}(\partial \Omega )}$ 

to the space ${\textstyle W^{1,p}(\Omega )}$ . By density of ${\textstyle C^{\infty }({\bar {\Omega }})}$  in ${\textstyle W^{1,p}(\Omega )}$  such an extension is possible if ${\textstyle T}$  is continuous with respect to the ${\textstyle W^{1,p}(\Omega )}$ -norm. The proof of this, i.e. that there exists ${\textstyle C>0}$  (depending on ${\textstyle \Omega }$  and ${\textstyle p}$ ) such that

${\displaystyle \|Tu\|_{L^{p}(\partial \Omega )}\leq C\|u\|_{W^{1,p}(\Omega )}}$  for all ${\displaystyle u\in C^{\infty }({\bar {\Omega }}).}$ 

is the central ingredient in the construction of the trace operator. A local variant of this estimate for ${\textstyle C^{1}({\bar {\Omega }})}$ -functions is first proven for a locally flat boundary using the divergence theorem. By transformation, a general ${\textstyle C^{1}}$ -boundary can be locally straightened to reduce to this case, where the ${\textstyle C^{1}}$ -regularity of the transformation requires that the local estimate holds for ${\textstyle C^{1}({\bar {\Omega }})}$ -functions.

With this continuity of the trace operator in ${\textstyle C^{\infty }({\bar {\Omega }})}$  an extension to ${\textstyle W^{1,p}(\Omega )}$  exists by abstract arguments and ${\textstyle Tu}$  for ${\textstyle u\in W^{1,p}(\Omega )}$  can be characterized as follows. Let ${\textstyle u_{k}\in C^{\infty }({\bar {\Omega }})}$  be a sequence approximating ${\textstyle u\in W^{1,p}(\Omega )}$  by density. By the proven continuity of ${\textstyle T}$  in ${\textstyle C^{\infty }({\bar {\Omega }})}$  the sequence ${\textstyle u_{k}|_{\partial \Omega }}$  is a Cauchy sequence in ${\textstyle L^{p}(\partial \Omega )}$  and ${\textstyle Tu=\lim _{k\to \infty }u_{k}|_{\partial \Omega }}$  with limit taken in ${\textstyle L^{p}(\partial \Omega )}$ .

The extension property ${\textstyle Tu=u|_{\partial \Omega }}$  holds for ${\textstyle u\in C^{\infty }({\bar {\Omega }})}$  by construction, but for any ${\textstyle u\in W^{1,p}(\Omega )\cap C({\bar {\Omega }})}$  there exists a sequence ${\textstyle u_{k}\in C^{\infty }({\bar {\Omega }})}$  which converges uniformly on ${\textstyle {\bar {\Omega }}}$  to ${\textstyle u}$ , verifying the extension property on the larger set ${\textstyle W^{1,p}(\Omega )\cap C({\bar {\Omega }})}$ .

The case p = ∞ edit

If ${\textstyle \Omega }$  is bounded and has a ${\textstyle C^{1}}$ -boundary then by Morrey's inequality there exists a continuous embedding ${\textstyle W^{1,\infty }(\Omega )\hookrightarrow C^{0,1}(\Omega )}$ , where ${\textstyle C^{0,1}(\Omega )}$  denotes the space of Lipschitz continuous functions. In particular, any function ${\textstyle u\in W^{1,\infty }(\Omega )}$  has a classical trace ${\textstyle u|_{\partial \Omega }\in C(\partial \Omega )}$  and there holds

${\displaystyle \|u|_{\partial \Omega }\|_{C(\partial \Omega )}\leq \|u\|_{C^{0,1}(\Omega )}\leq C\|u\|_{W^{1,\infty }(\Omega )}.}$ 

The Sobolev spaces ${\textstyle W_{0}^{1,p}(\Omega )}$  for ${\textstyle 1\leq p<\infty }$  are defined as the closure of the set of compactly supported test functions ${\textstyle C_{c}^{\infty }(\Omega )}$  with respect to the ${\textstyle W^{1,p}(\Omega )}$ -norm. The following alternative characterization holds:

${\displaystyle W_{0}^{1,p}(\Omega )=\{u\in W^{1,p}(\Omega )\mid Tu=0\}=\ker(T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega )),}$ 

where ${\textstyle \ker(T)}$  is the kernel of ${\textstyle T}$ , i.e. ${\textstyle W_{0}^{1,p}(\Omega )}$  is the subspace of functions in ${\textstyle W^{1,p}(\Omega )}$  with trace zero.

For p > 1 edit

The trace operator is not surjective onto ${\textstyle L^{p}(\partial \Omega )}$  if ${\textstyle p>1}$ , i.e. not every function in ${\textstyle L^{p}(\partial \Omega )}$  is the trace of a function in ${\textstyle W^{1,p}(\Omega )}$ . As elaborated below the image consists of functions which satisfy an ${\textstyle L^{p}}$ -version of Hölder continuity.

Abstract characterization edit

An abstract characterization of the image of ${\textstyle T}$  can be derived as follows. By the isomorphism theorems there holds

${\displaystyle T(W^{1,p}(\Omega ))\cong W^{1,p}(\Omega )/\ker(T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega ))=W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )}$ 

where ${\textstyle X/N}$  denotes the quotient space of the Banach space ${\textstyle X}$  by the subspace ${\textstyle N\subset X}$  and the last identity follows from the characterization of ${\textstyle W_{0}^{1,p}(\Omega )}$  from above. Equipping the quotient space with the quotient norm defined by

${\displaystyle \|u\|_{W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )}=\inf _{u_{0}\in W_{0}^{1,p}(\Omega )}\|u-u_{0}\|_{W^{1,p}(\Omega )}}$ 

the trace operator ${\textstyle T}$  is then a surjective, bounded linear operator

${\displaystyle T\colon W^{1,p}(\Omega )\to W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )}$ .

Characterization using Sobolev–Slobodeckij spaces edit

A more concrete representation of the image of ${\textstyle T}$  can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the ${\textstyle L^{p}}$ -setting. Since ${\textstyle \partial \Omega }$  is a (n-1)-dimensional Lipschitz manifold embedded into ${\textstyle \mathbb {R} ^{n}}$  an explicit characterization of these spaces is technically involved. For simplicity consider first a planar domain ${\textstyle \Omega '\subset \mathbb {R} ^{n-1}}$ . For ${\textstyle v\in L^{p}(\Omega ')}$  define the (possibly infinite) norm

${\displaystyle \|v\|_{W^{1-1/p,p}(\Omega ')}=\left(\|v\|_{L^{p}(\Omega ')}^{p}+\int _{\Omega '\times \Omega '}{\frac {|v(x)-v(y)|^{p}}{|x-y|^{(1-1/p)p+(n-1)}}}\,\mathrm {d} (x,y)\right)^{1/p}}$ 

which generalizes the Hölder condition ${\textstyle |v(x)-v(y)|\leq C|x-y|^{1-1/p}}$ . Then

${\displaystyle W^{1-1/p,p}(\Omega ')=\left\{v\in L^{p}(\Omega ')\;\mid \;\|v\|_{W^{1-1/p,p}(\Omega ')}<\infty \right\}}$ 

equipped with the previous norm is a Banach space (a general definition of ${\textstyle W^{s,p}(\Omega ')}$  for non-integer ${\textstyle s>0}$  can be found in the article for Sobolev-Slobodeckij spaces). For the (n-1)-dimensional Lipschitz manifold ${\textstyle \partial \Omega }$  define ${\textstyle W^{1-1/p,p}(\partial \Omega )}$  by locally straightening ${\textstyle \partial \Omega }$  and proceeding as in the definition of ${\textstyle W^{1-1/p,p}(\Omega ')}$ .

The space ${\textstyle W^{1-1/p,p}(\partial \Omega )}$  can then be identified as the image of the trace operator and there holds^[1] that

${\displaystyle T\colon W^{1,p}(\Omega )\to W^{1-1/p,p}(\partial \Omega )}$ 

is a surjective, bounded linear operator.

For p = 1 edit

For ${\textstyle p=1}$  the image of the trace operator is ${\textstyle L^{1}(\partial \Omega )}$  and there holds^[1] that

${\displaystyle T\colon W^{1,1}(\Omega )\to L^{1}(\partial \Omega )}$ 

is a surjective, bounded linear operator.

The trace operator is not injective since multiple functions in ${\textstyle W^{1,p}(\Omega )}$  can have the same trace (or equivalently, ${\textstyle W_{0}^{1,p}(\Omega )\neq 0}$ ). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for ${\textstyle 1<p<\infty }$  there exists a bounded, linear trace extension operator^[3]

${\displaystyle E\colon W^{1-1/p,p}(\partial \Omega )\to W^{1,p}(\Omega )}$ ,

using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that

${\displaystyle T(Ev)=v}$  for all ${\textstyle v\in W^{1-1/p,p}(\partial \Omega )}$ 

and, by continuity, there exists ${\textstyle C>0}$  with

${\displaystyle \|Ev\|_{W^{1,p}(\Omega )}\leq C\|v\|_{W^{1-1/p,p}(\partial \Omega )}}$ .

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 ${\textstyle W^{1,p}(\Omega )\to W^{1,p}(\mathbb {R} ^{n})}$  which play a fundamental role in the theory of Sobolev spaces.

Higher derivatives edit

Many of the previous results can be extended to ${\textstyle W^{m,p}(\Omega )}$  with higher differentiability ${\textstyle m=2,3,\ldots }$  if the domain is sufficiently regular. Let ${\textstyle N}$  denote the exterior unit normal field on ${\textstyle \partial \Omega }$ . Since ${\textstyle u|_{\partial \Omega }}$  can encode differentiability properties in tangential direction only the normal derivative ${\textstyle \partial _{N}u|_{\partial \Omega }}$  is of additional interest for the trace theory for ${\textstyle m=2}$ . Similar arguments apply to higher-order derivatives for ${\textstyle m>2}$ .

Let ${\textstyle 1<p<\infty }$  and ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  be a bounded domain with ${\textstyle C^{m,1}}$ -boundary. Then^[3] there exists a surjective, bounded linear higher-order trace operator

${\displaystyle T_{m}\colon W^{m,p}(\Omega )\to \prod _{l=0}^{m-1}W^{m-l-1/p,p}(\partial \Omega )}$ 

with Sobolev-Slobodeckij spaces ${\textstyle W^{s,p}(\partial \Omega )}$  for non-integer ${\textstyle s>0}$  defined on ${\textstyle \partial \Omega }$  through transformation to the planar case ${\textstyle W^{s,p}(\Omega ')}$  for ${\textstyle \Omega '\subset \mathbb {R} ^{n-1}}$ , whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator ${\textstyle T_{m}}$  extends the classical normal traces in the sense that

${\displaystyle T_{m}u=\left(u|_{\partial \Omega },\partial _{N}u|_{\partial \Omega },\ldots ,\partial _{N}^{m-1}u|_{\partial \Omega }\right)}$  for all ${\textstyle u\in W^{m,p}(\Omega )\cap C^{m-1}({\bar {\Omega }}).}$ 

Furthermore, there exists a bounded, linear right-inverse of ${\textstyle T_{m}}$ , a higher-order trace extension operator^[3]

${\displaystyle E_{m}\colon \prod _{l=0}^{m-1}W^{m-l-1/p,p}(\partial \Omega )\to W^{m,p}(\Omega )}$ .

Finally, the spaces ${\textstyle W_{0}^{m,p}(\Omega )}$ , the completion of ${\textstyle C_{c}^{\infty }(\Omega )}$  in the ${\textstyle W^{m,p}(\Omega )}$ -norm, can be characterized as the kernel of ${\textstyle T_{m}}$ ,^[3] i.e.

${\displaystyle W_{0}^{m,p}(\Omega )=\{u\in W^{m,p}(\Omega )\mid T_{m}u=0\}}$ .

Less regular spaces edit

No trace in L^p edit

There is no sensible extension of the concept of traces to ${\textstyle L^{p}(\Omega )}$  for ${\textstyle 1\leq p<\infty }$  since any bounded linear operator which extends the classical trace must be zero on the space of test functions ${\textstyle C_{c}^{\infty }(\Omega )}$ , which is a dense subset of ${\textstyle L^{p}(\Omega )}$ , implying that such an operator would be zero everywhere.

Generalized normal trace edit

Let ${\textstyle \operatorname {div} v}$  denote the distributional divergence of a vector field ${\textstyle v}$ . For ${\textstyle 1<p<\infty }$  and bounded Lipschitz domain ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  define

${\displaystyle E_{p}(\Omega )=\{v\in (L^{p}(\Omega ))^{n}\mid \operatorname {div} v\in L^{p}(\Omega )\}}$ 

which is a Banach space with norm

${\displaystyle \|v\|_{E_{p}(\Omega )}=\left(\|v\|_{L^{p}(\Omega )}^{p}+\|\operatorname {div} v\|_{L^{p}(\Omega )}^{p}\right)^{1/p}}$ .

Let ${\textstyle N}$  denote the exterior unit normal field on ${\textstyle \partial \Omega }$ . Then^[4] there exists a bounded linear operator

${\displaystyle T_{N}\colon E_{p}(\Omega )\to (W^{1-1/q,q}(\partial \Omega ))'}$ ,

where ${\textstyle q=p/(p-1)}$  is the conjugate exponent to ${\textstyle p}$  and ${\textstyle X'}$  denotes the continuous dual space to a Banach space ${\textstyle X}$ , such that ${\textstyle T_{N}}$  extends the normal trace ${\textstyle (v\cdot N)|_{\partial \Omega }}$  for ${\textstyle v\in (C^{\infty }({\bar {\Omega }}))^{n}}$  in the sense that

${\displaystyle T_{N}v={\bigl \{}\varphi \in W^{1-1/q,q}(\partial \Omega )\mapsto \int _{\partial \Omega }\varphi v\cdot N\,\mathrm {d} S{\bigr \}}}$ .

The value of the normal trace operator ${\textstyle (T_{N}v)(\varphi )}$  for ${\textstyle \varphi \in W^{1-1/q,q}(\partial \Omega )}$  is defined by application of the divergence theorem to the vector field ${\textstyle w=E\varphi \,v}$  where ${\textstyle E}$  is the trace extension operator from above.

Application. Any weak solution ${\textstyle u\in H^{1}(\Omega )}$  to ${\textstyle -\Delta u=f\in L^{2}(\Omega )}$  in a bounded Lipschitz domain ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  has a normal derivative in the sense of ${\textstyle T_{N}\nabla u\in (W^{1/2,2}(\partial \Omega ))^{*}}$ . This follows as ${\textstyle \nabla u\in E_{2}(\Omega )}$  since ${\textstyle \nabla u\in L^{2}(\Omega )}$  and ${\textstyle \operatorname {div} (\nabla u)=\Delta u=-f\in L^{2}(\Omega )}$ . This result is notable since in Lipschitz domains in general ${\textstyle u\not \in H^{2}(\Omega )}$ , such that ${\textstyle \nabla u}$  may not lie in the domain of the trace operator ${\textstyle T}$ .

The theorems presented above allow a closer investigation of the boundary value problem

${\displaystyle {\begin{alignedat}{2}-\Delta u&=f&\quad &{\text{in }}\Omega ,\\u&=g&&{\text{on }}\partial \Omega \end{alignedat}}}$ 

on a Lipschitz domain ${\textstyle \Omega \subset \mathbb {R} ^{n}}$  from the motivation. Since only the Hilbert space case ${\textstyle p=2}$  is investigated here, the notation ${\textstyle H^{1}(\Omega )}$  is used to denote ${\textstyle W^{1,2}(\Omega )}$  etc. As stated in the motivation, a weak solution ${\textstyle u\in H^{1}(\Omega )}$  to this equation must satisfy ${\textstyle Tu=g}$  and

${\displaystyle \int _{\Omega }\nabla u\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x}$  for all ${\textstyle \varphi \in H_{0}^{1}(\Omega )}$ ,

where the right-hand side must be interpreted for ${\textstyle f\in H^{-1}(\Omega )=(H_{0}^{1}(\Omega ))'}$  as a duality product with the value ${\textstyle f(\varphi )}$ .

Existence and uniqueness of weak solutions edit

The characterization of the range of ${\textstyle T}$  implies that for ${\textstyle Tu=g}$  to hold the regularity ${\textstyle g\in H^{1/2}(\partial \Omega )}$  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 ${\textstyle Eg\in H^{1}(\Omega )}$  such that ${\textstyle T(Eg)=g}$ . Defining ${\textstyle u_{0}}$  by ${\textstyle u_{0}=u-Eg}$  we have that ${\textstyle Tu_{0}=Tu-T(Eg)=0}$  and thus ${\textstyle u_{0}\in H_{0}^{1}(\Omega )}$  by the characterization of ${\textstyle H_{0}^{1}(\Omega )}$  as space of trace zero. The function ${\textstyle u_{0}\in H_{0}^{1}(\Omega )}$  then satisfies the integral equation

${\displaystyle \int _{\Omega }\nabla u_{0}\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }\nabla (u-Eg)\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x-\int _{\Omega }\nabla Eg\cdot \nabla \varphi \,\mathrm {d} x}$  for all ${\textstyle \varphi \in H_{0}^{1}(\Omega )}$ .

Thus the problem with inhomogeneous boundary values for ${\textstyle u}$  could be reduced to a problem with homogeneous boundary values for ${\textstyle u_{0}}$ , a technique which can be applied to any linear differential equation. By the Riesz representation theorem there exists a unique solution ${\textstyle u_{0}}$  to this problem. By uniqueness of the decomposition ${\textstyle u=u_{0}+Eg}$ , this is equivalent to the existence of a unique weak solution ${\textstyle u}$  to the inhomogeneous boundary value problem.

Continuous dependence on the data edit

It remains to investigate the dependence of ${\textstyle u}$  on ${\textstyle f}$  and ${\textstyle g}$ . Let ${\textstyle c_{1},c_{2},\ldots >0}$  denote constants independent of ${\textstyle f}$  and ${\textstyle g}$ . By continuous dependence of ${\textstyle u_{0}}$  on the right-hand side of its integral equation, there holds

${\displaystyle \|u_{0}\|_{H_{0}^{1}(\Omega )}\leq c_{1}\left(\|f\|_{H^{-1}(\Omega )}+\|Eg\|_{H^{1}(\Omega )}\right)}$ 

and thus, using that ${\textstyle \|u_{0}\|_{H_{0}^{1}(\Omega )}\leq c_{2}\|u_{0}\|_{H^{1}(\Omega )}}$  and ${\textstyle \|Eg\|_{H^{1}(\Omega )}\leq c_{3}\|g\|_{H^{1/2}(\Omega )}}$  by continuity of the trace extension operator, it follows that

${\displaystyle {\begin{aligned}\|u\|_{H^{1}(\Omega )}&\leq \|u_{0}\|_{H^{1}(\Omega )}+\|Eg\|_{H^{1}(\Omega )}\leq c_{1}c_{2}\|f\|_{H^{-1}(\Omega )}+(1+c_{1}c_{2})\|Eg\|_{H^{1}(\Omega )}\\&\leq c_{4}\left(\|f\|_{H^{-1}(\Omega )}+\|g\|_{H^{1/2}(\partial \Omega )}\right)\end{aligned}}}$ 

and the solution map

${\displaystyle H^{-1}(\Omega )\times H^{1/2}(\partial \Omega )\ni (f,g)\mapsto u\in H^{1}(\Omega )}$ 

is therefore continuous.

• Trace class
• Nuclear operators between Banach spaces

• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8