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Concept
Trace operator
Summary
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.
On a bounded, smooth domain , consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:
with given functions and with regularity discussed in the application section below. The weak solution of this equation must satisfy
for all .
The -regularity of is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense can satisfy the boundary condition on : by definition, is an equivalence class of functions which can have arbitrary values on since this is a null set with respect to the n-dimensional Lebesgue measure.
If there holds by Sobolev's embedding theorem, such that can satisfy the boundary condition in the classical sense, i.e. the restriction of to agrees with the function (more precisely: there exists a representative of in with this property). For with such an embedding does not exist and the trace operator presented here must be used to give meaning to . Then with 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 for sufficiently regular .
The trace operator can be defined for functions in the Sobolev spaces with , see the section below for possible extensions of the trace to other spaces. Let for be a bounded domain with Lipschitz boundary. Then there exists a bounded linear trace operator
such that extends the classical trace, i.e.
for all .
The continuity of implies that
for all
with constant only depending on and . The function is called trace of and is often simply denoted by .
Official source
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