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Wikipedia / Trace operator

In mathematical analysis, 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, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:

with regularity discussed in the application section of this equation must satisfy

  1. .

is sufficient for the well-definedness of this integral equation is an equivalence class since this is a null set with respect to the n-dimensional Lebesgue measure.

by Sobolev's embedding theorem.

Trace theorem

The trace operator can be defined for functions in the Sobolev spaces be a bounded domain with Lipschitz boundary. Then there exists a bounded linear trace operator

extends the classical trace, i.e.

  1. .

implies that

.

Construction

This paragraph follows Evans,-boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.-domain, the trace operator can be defined as continuous linear extension of the operator

. By density) such that

  1. ,

-functions is first proven for a locally flat boundary using the divergence theorem-functions.

is a Cauchy sequence.

.

The case p = ∞

-boundary then by Morrey's inequality denotes the space of Lipschitz continuous and there holds

Functions with trace zero

are defined as the closure-norm. The following alternative characterization holds:

is the kernel with trace zero.

Image of the trace operator

For p > 1

-version of Hölder continuity.

Abstract characterization

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

denotes the quotient space from above. Equipping the quotient space with the quotient norm defined by

is then a surjective, bounded linear operator

  1. .

Characterization using Sobolev–Slobodeckij spaces

can be given using Sobolev-Slobodeckij spaces is a (n-1)-dimensional Lipschitz manifold define the (possibly infinite) norm

. Then

can be found in the article for Sobolev-Slobodeckij spaces). For the (n-1).

can then be identified as the image of the trace operator and there holds that

is a surjective, bounded linear operator.

For p = 1

and there holds that

is a surjective, bounded linear operator.

Right-inverse: trace extension operator

there exists a bounded, linear trace extension operator

  1. ,

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

with

  1. .

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 which play a fundamental role in the theory of Sobolev spaces.

Extension to other spaces

Higher derivatives

.

-boundary. Then there exists a surjective, bounded linear higher-order trace operator

, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces extends the classical normal traces in the sense that

, a higher-order trace extension operator

  1. .

, i.e.

  1. .

Less regular spaces

No trace in Lp

, implying that such an operator would be zero everywhere.

Generalized normal trace

denote the distributional divergence of a vector field define

which is a Banach space with norm

  1. .

. Then there exists a bounded linear operator

  1. ,

is the conjugate exponent denotes the continuous dual space in the sense that

  1. .

is defined by application of the divergence theorem is the trace extension operator from above.

Application..

Application

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

from the motivation. Since only the Hilbert space and

  1. ,

.

Existence and uniqueness of weak solutions

then satisfies the integral equation

  1. .

, a technique which can be applied to any linear differential equation. By the Riesz representation theorem to the inhomogeneous boundary value problem.

Continuous dependence on the data

on the right-hand side of its integral equation, there holds

by continuity of the trace extension operator, it follows that

and the solution map

is therefore continuous.

See also

References

de:Sobolev-Raum#Spuroperator