Traces of Functions on Sobolev Spaces

Abstract

It follows from the “regularity” of the Sobolev spaces that we can “extend” a function belonging to W 1,p (Ω) to the boundary when the open set Ω is sufficiently regular. This extension is needed to define the Dirichlet problems studied further on. A first step allows us to define the trace as a function in L p (∂Ω), and a more detailed study lets us show that these traces have more regularity, which moreover characterizes them. We therefore introduce the fractional derivative and the so-called fractional Sobolev spaces, where the degree of differentiation is not integral. For example, the trace of a function in W 1,2(Ω) is “one half” differentiable. As in the case of Sobolev spaces with integral differentiation degree, the fractional Sobolev spaces have density properties for the regular functions, embeddings into “more regular” spaces, and compactness results when the “boundary” is bounded.