The variable exponent BV-Sobolev capacity

TL;DR

This paper investigates the properties of a variable exponent BV-Sobolev capacity, providing a new definition of the mixed BV-Sobolev space, and establishing its capacity as a Choquet capacity with key properties.

Contribution

It introduces an alternative, relaxation-based definition of the mixed BV-Sobolev space for variable exponents, and analyzes the resulting capacity's properties and null sets.

Findings

The new definition produces a Banach space coinciding with the original for bounded domains.

The capacity is shown to be a Choquet capacity with standard properties.

The capacity shares null sets with the variable exponent Sobolev capacity under log-Hölder continuity.

Abstract

In this article we study basic properties of the mixed BV-Sobolev capacity with variable exponent p. We give an alternative way to define mixed type BV-Sobolev-space which was originally introduced by Harjulehto, H\"ast\"o, and Latvala. Our definition is based on relaxing the p-energy functional with respect to the Lebesgue space topology. We prove that this procedure produces a Banach space that coincides with the space defined by Harjulehto et al. for bounded domain and log-H\"older continuous exponent p. Then we show that this induces a type of variable exponent BV-capacity and that this is a Choquet capacity with many usual properties. Finally, we prove that this capacity has the same null sets as the variable exponent Sobolev capacity when p is log-H\"older continuous.