Sobolev functions whose inner trace at the boundary is zero

Abstract

Let Ω⊂Rⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev function f∈W^{1, p}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then f is weakly zero on ϖΩ in the sense that f∈W ${}_{0}^{1,p}$ (Ω).