Trace operator

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 by 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 ${\textstyle W^{1,p}(\Omega )\cap C({\bar {\Omega }})}$.

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 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 a ${\textstyle L^{p}}$-version of Hölder continuity.

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

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

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 ${\textstyle p=1}$ 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.

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.

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

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.

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 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 ${\textstyle u\in H_{0}^{1}(\Omega )}$ 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.

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.