Revista Matemática Complutense

, Volume 27, Issue 1, pp 13–40

The variable exponent BV-Sobolev capacity


DOI: 10.1007/s13163-012-0109-8

Cite this article as:
Hakkarainen, H. & Nuortio, M. Rev Mat Complut (2014) 27: 13. doi:10.1007/s13163-012-0109-8


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.


CapacityFunctions of bounded variationSobolev spacesVariable exponent

Mathematics Subject Classification (2010)


Copyright information

© Universidad Complutense de Madrid 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of OuluFinland