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 $\Omega \subset \mathbb R^n$, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:

:$\begin -\Delta u &= f &\quad&\text \Omega,\\ u &= g &&\text \partial \Omega \end$

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

:$\int_\Omega \nabla u \cdot \nabla \varphi \,\mathrm dx = \int_\Omega f \varphi \,\mathrm dx$ for all $\varphi \in H^1_0(\Omega)$.

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

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

Trace theorem

The trace operator can be defined for functions in the Sobolev spaces $W^(\Omega)$ with $1 \leq p < \infty$, see the section below for possible extensions of the trace to other spaces. Let $\Omega \subset \mathbb R^n$ for $n \in \mathbb N$ be a bounded domain with Lipschitz boundary. Then there exists a bounded linear trace operator
: $T\colon W^(\Omega) \to L^p(\partial \Omega)$
such that $T$ extends the classical trace, i.e.
: $T u = u , _$ for all $u \in W^(\Omega) \cap C(\bar \Omega)$.
The continuity of $T$ implies that
: $\, T u \, _ \leq C \, u \, _$ for all $u \in W^(\Omega)$
with constant only depending on $p$ and $\Omega$. The function $T u$ is called trace of $u$ and is often simply denoted by $u , _$. Other common symbols for $T$ include $tr$ and $\gamma$.

Construction

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

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

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

: $\, Tu\, _\le C \, u\, _$ for all $u \in C^\infty(\bar \Omega).$

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

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

The extension property $T u = u , _$ holds for $u \in C^(\bar \Omega)$ by construction, but for any $u \in W^(\Omega) \cap C(\bar \Omega)$ there exists a sequence $u_k \in C^\infty(\bar \Omega)$ which converges uniformly on $\bar \Omega$ to $u$, verifying the extension property on the larger set $W^(\Omega) \cap C(\bar \Omega)$.

The case p = ∞

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

: $\, u , _ \, _ \leq \, u \, _ \leq C \, u \, _.$

Functions with trace zero

The Sobolev spaces $W^_0(\Omega)$ for $1 \leq p < \infty$ are defined as the closure of the set of compactly supported test functions $C^\infty_c(\Omega)$ with respect to the $W^(\Omega)$-norm. The following alternative characterization holds:

: $W^_0(\Omega) = \ = \ker(T\colon W^(\Omega) \to L^p(\partial \Omega)),$

where $\ker(T)$ is the kernel of $T$, i.e. $W^_0(\Omega)$ is the subspace of functions in $W^(\Omega)$ with trace zero.

Image of the trace operator

For p > 1

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

Abstract characterization

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

: $T(W^(\Omega)) \cong W^(\Omega) / \ker(T\colon W^(\Omega) \to L^p(\partial \Omega)) = W^(\Omega) / W^_0(\Omega)$

where $X / N$ denotes the quotient space of the Banach space $X$ by the subspace $N \subset X$ and the last identity follows from the characterization of $W^_0(\Omega)$ from above. Equipping the quotient space with the quotient norm defined by

: $\, u\, _ = \inf_ \, u - u_0\, _$

the trace operator $T$ is then a surjective, bounded linear operator

: $T\colon W^(\Omega) \to W^(\Omega) / W^_0(\Omega)$.

Characterization using Sobolev–Slobodeckij spaces

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

: $\, v \, _ = \left( \, v\, _^p + \int_ \frac\,\mathrm d(x, y) \right)^$
:
which generalizes the Hölder condition $, v(x) - v(y) , \leq C , x - y, ^$. Then

: $W^(\Omega') = \left\$

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

The space $W^(\partial \Omega)$ can then be identified as the image of the trace operator and there holds that

: $T\colon W^(\Omega) \to W^(\partial \Omega)$

is a surjective, bounded linear operator.

For p = 1

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

: $T\colon W^(\Omega) \to L^1(\partial \Omega)$

is a surjective, bounded linear operator.

Right-inverse: trace extension operator

The trace operator is not injective since multiple functions in $W^(\Omega)$ can have the same trace (or equivalently, $W^_0(\Omega) \neq 0$). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for $1 < p < \infty$ there exists a bounded, linear trace extension operator

: $E\colon W^(\partial \Omega) \to W^(\Omega)$,

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

: $T (E v) = v$ for all $v \in W^(\partial \Omega)$

and, by continuity, there exists $C > 0$ with

: $\, E v \, _ \leq C \, v \, _$.

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

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

: $T_m\colon W^(\Omega) \to \prod_^ W^(\partial \Omega)$
:
with Sobolev-Slobodeckij spaces $W^(\partial \Omega)$ for non-integer $s > 0$ defined on $\partial \Omega$ through transformation to the planar case $W^(\Omega')$ for $\Omega' \subset \mathbb R^$, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator $T_m$ extends the classical normal traces in the sense that

: $T_m u = \left(u , _, \partial_N u , _, \ldots, \partial_N^ u , _\right)$ for all $u \in W^(\Omega) \cap C^(\bar \Omega).$

Furthermore, there exists a bounded, linear right-inverse of $T_m$, a higher-order trace extension operator

: $E_m\colon \prod_^ W^(\partial \Omega) \to W^(\Omega)$.

Finally, the spaces $W^_0(\Omega)$, the completion of $C^\infty_c(\Omega)$ in the $W^(\Omega)$-norm, can be characterized as the kernel of $T_m$, i.e.

: $W^_0(\Omega) = \$.

Less regular spaces

No trace in ''L^p''

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

Generalized normal trace

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

:$E_p(\Omega) = \$

which is a Banach space with norm

:$\, v \, _ = \left( \, v \, _^p + \, \operatorname v \, _^p \right)^$.

Let $N$ denote the exterior unit normal field on $\partial \Omega$. Then there exists a bounded linear operator

: $T_N\colon E_p(\Omega) \to (W^(\partial \Omega))'$,

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

: $T_N v = \bigl\$.

The value of the normal trace operator $(T_N v)(\varphi)$ for $\varphi \in W^(\partial \Omega)$ is defined by application of the divergence theorem to the vector field $w = E \varphi \, v$ where $E$ is the trace extension operator from above.

''Application.'' Any weak solution $u \in H^1(\Omega)$ to $- \Delta u = f \in L^2(\Omega)$ in a bounded Lipschitz domain $\Omega \subset \mathbb R^n$ has a normal derivative in the sense of $T_N \nabla u \in (W^(\partial \Omega))^*$. This follows as $\nabla u \in E_2(\Omega)$ since $\nabla u \in L^2(\Omega)$ and $\operatorname(\nabla u) = \Delta u = - f \in L^2(\Omega)$. This result is notable since in Lipschitz domains in general $u \not\in H^2(\Omega)$, such that $\nabla u$ may not lie in the domain of the trace operator $T$.

Application

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

:$\begin -\Delta u &= f &\quad&\text \Omega,\\ u &= g &&\text \partial \Omega \end$

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

:$\int_\Omega \nabla u \cdot \nabla \varphi \,\mathrm dx = \int_\Omega f \varphi \,\mathrm dx$ for all $\varphi \in H^1_0(\Omega)$,

where the right-hand side must be interpreted for $f \in H^(\Omega) = (H^1_0(\Omega))'$ as a duality product with the value $f(\varphi)$.

Existence and uniqueness of weak solutions

The characterization of the range of $T$ implies that for $T u = g$ to hold the regularity $g \in H^(\partial \Omega)$ 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 $Eg \in H^1(\Omega)$ such that $T(Eg) = g$. Defining $u_0$ by $u_0 = u - Eg$ we have that $T u_0 = Tu - T(Eg) = 0$ and thus $u_0 \in H^1_0(\Omega)$ by the characterization of $H^1_0(\Omega)$ as space of trace zero. The function $u_0 \in H^1_0(\Omega)$ then satisfies the integral equation

:$\int_\Omega \nabla u_0 \cdot \nabla \varphi \,\mathrm dx = \int_\Omega \nabla (u - Eg) \cdot \nabla \varphi \, \mathrm dx = \int_\Omega f \varphi \,\mathrm dx - \int_\Omega \nabla Eg \cdot \nabla \varphi \,\mathrm dx$ for all $\varphi \in H^1_0(\Omega)$.

Thus the problem with inhomogeneous boundary values for $u$ could be reduced to a problem with homogeneous boundary values for $u_0$, a technique which can be applied to any linear differential equation. By the Riesz representation theorem there exists a unique solution $u_0$ to this problem. By uniqueness of the decomposition $u = u_0 + Eg$, this is equivalent to the existence of a unique weak solution $u$ to the inhomogeneous boundary value problem.

Continuous dependence on the data

It remains to investigate the dependence of $u$ on $f$ and $g$. Let $c_1, c_2, \ldots > 0$ denote constants independent of $f$ and $g$. By continuous dependence of $u_0$ on the right-hand side of its integral equation, there holds

: $\, u_0 \, _ \leq c_1 \left( \, f\, _ + \, Eg\, _ \right)$

and thus, using that $\, u_0 \, _ \leq c_2 \, u_0 \, _$ and $\, E g \, _ \leq c_3 \, g \, _$ by continuity of the trace extension operator, it follows that

: $\begin\, u \, _ &\leq \, u_0 \, _ + \, Eg \, _ \leq c_1 c_2 \, f\, _ + (1+c_1 c_2) \, Eg\, _ \\ &\leq c_4 \left(\, f\, _ + \, g\, _ \right)\end$

and the solution map

: $H^(\Omega) \times H^(\partial \Omega) \ni (f, g) \mapsto u \in H^1(\Omega)$

is therefore continuous.

See also

* Trace class
* Nuclear operators between Banach spaces

References

*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

Sobolev spaces
Operator theory

de:Sobolev-Raum#Spuroperator