, Volume 27, Issue 1, pp 13-40
Date: 04 Dec 2012

The variable exponent BV-Sobolev capacity

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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 the mixed type BV-Sobolev-space which was originally introduced by Harjulehto, Hästö, 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 a bounded domain \(\Omega \) and a log-Hölder 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 if \(p\) is log-Hölder continuous, then this capacity has the same null sets as the variable exponent Sobolev capacity.