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.

, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions: with given functions

since this is a null set with respect to the n-dimensional Lebesgue measure.

can satisfy the boundary condition in the classical sense, i.e. the restriction of

is called a weak solution to the boundary value problem if the integral equation above is satisfied.

The trace operator can be defined for functions in the Sobolev spaces

A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.

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

are defined as the closure of the set of compactly supported test functions

is then a surjective, bounded linear operator A more concrete representation of the image of

can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the

define the (possibly infinite) norm which generalizes the Hölder condition

Then equipped with the previous norm is a Banach space (a general definition of

The trace operator is not injective since multiple functions in

The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain.

there exists a bounded, linear trace extension operator[3] using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that and, by continuity, there exists

with Notable is not the mere existence but the linearity and continuity of the right inverse.

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

can encode differentiability properties in tangential direction only the normal derivative

Then[3] there exists a surjective, bounded linear higher-order trace operator with Sobolev-Slobodeckij spaces

, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces.

extends the classical normal traces in the sense that Furthermore, there exists a bounded, linear right-inverse of

, a higher-order trace extension operator[3] Finally, the spaces

since any bounded linear operator which extends the classical trace must be zero on the space of test functions

is defined by application of the divergence theorem to the vector field

This result is notable since in Lipschitz domains in general

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

then satisfies the integral equation Thus the problem with inhomogeneous boundary values for

By the Riesz representation theorem there exists a unique solution