A Note on the Equivalence and the Boundary Behavior of a Class of Sobolev Capacities. Issue 3 (16th April 2018)

• Record Type: Journal Article
• Authors: Christof, Constantin; Müller, Georg
• Is Part Of: Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik. Volume 40:Issue 3(2017)
• Page Start: 238
• Page End: 266
• Publication Date: 2018-04-16
• Subjects: capacity theory -- boundary -- optimal control -- Signorini -- variational inequalities -- contact mechanics -- sensitivity analysis -- classification problems; Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; 510.5
• DOI: 10.1002/gamm.201730005
• Languages: English
• ISSNs: 0936-7195

Abstract

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 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.