Fractional variable exponents Sobolev trace spaces

TL;DR

This paper introduces fractional variable exponent Sobolev trace spaces, characterizes zero trace spaces via capacity, and discusses removable sets, advancing the understanding of variable exponent Sobolev spaces on open sets.

Contribution

It develops the theory of fractional variable exponent Sobolev trace spaces, including quasicontinuous representatives and capacity-based characterizations, which are novel contributions.

Findings

Every Sobolev function class has a quasicontinuous representative

Zero trace spaces are characterized by relative capacity

Removable sets are identified via capacity criteria

Abstract

We introduce and study fractional variable exponents Sobolev trace spaces on any open set in the Euclidean space equipped with the Lebesgue measure. We show that every equivalence class of Sobolev functions has a quasicontinuous representatives. We use the relative capacity to characterize completely the zero trace fractional variable exponents Sobolev spaces. We also give a relative capacity criterium for removable sets.