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.
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Acknowledgments
The authors thank Peter Hästö and Juha Kinnunen for discussions and suggestions. The authors also thank Tomasz Adamowicz, Teemu Lukkari, and Mikko Parviainen for reading earlier versions of the manuscript and suggestions. The authors thank the referee of Rev. Mat. Complut. for valuable remarks. The first author was partially supported by the Väisälä foundation of Suomalainen tiedeakatemia. The second author was partially supported by the Magnus Ehrnrooth foundation of Suomen tiedeseura, and partially by the Academy of Finland, project leader Peter Hästö, project in nonlinear partial differential equations.
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Hakkarainen, H., Nuortio, M. The variable exponent BV-Sobolev capacity. Rev Mat Complut 27, 13–40 (2014). https://doi.org/10.1007/s13163-012-0109-8
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DOI: https://doi.org/10.1007/s13163-012-0109-8