A Note on the Equivalence and the Boundary Behavior of a Class of Sobolev Capacities

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Christof, Constantin ; Müller, Georg:
A Note on the Equivalence and the Boundary Behavior of a Class of Sobolev Capacities.
Bayreuth , 2017 . - 27 S.

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Abstract

The purpose of this paper is to study different notions of Sobolev capacity commonly
used in the analysis of obstacle and Signorini type variational inequalities. We review basic facts from
nonlinear potential theory in an abstract setting that is tailored to the study of W^{1,p}- and W^{1−1/p,p} - capacities and we prove equivalency results that relate several approaches found in the literature to each other. Motivated by applications in contact mechanics, we especially focus on the behavior of different Sobolev capacities on and near the boundary of the domain in question. As a result, we obtain, for example, that the most common approaches to the sensitivity analysis of Signorini type problems are exactly the same.