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A Note on the Equivalence and the Boundary Behavior of a Class of Sobolev Capacities
Author(s) -
Christof Constantin,
Müller Georg
Publication year - 2018
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201730005
Subject(s) - sobolev space , mathematics , equivalence (formal languages) , focus (optics) , boundary (topology) , class (philosophy) , variational inequality , obstacle , domain (mathematical analysis) , type (biology) , mathematical analysis , pure mathematics , computer science , artificial intelligence , physics , ecology , law , political science , optics , biology
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 capacity 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.