2018
DOI: 10.1002/gamm.201730005
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A Note on the Equivalence and the Boundary Behavior of a Class of Sobolev Capacities

Abstract: MSC (2000) 28C15, 31B15, 31B25, 31C15, 49K40, 74M15The 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 fo… Show more

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Cited by 5 publications
(4 citation statements)
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“…P r o o f. It is well known that the spaces H s 0 (Ω), s ∈ (0, 1], and H 1/2 (∂Ω) are regular Dirichlet spaces, cf. [18] and also [19,Remark 3.2(c)]. The existence of sets with zero measure but non-zero capacity in these spaces follows, e.g., from [17, Theorem 5.5.1] (for H 1/2 (∂Ω), we can use a rectification argument here).…”
Section: Wwwgamm-mitteilungenorgmentioning
confidence: 92%
“…P r o o f. It is well known that the spaces H s 0 (Ω), s ∈ (0, 1], and H 1/2 (∂Ω) are regular Dirichlet spaces, cf. [18] and also [19,Remark 3.2(c)]. The existence of sets with zero measure but non-zero capacity in these spaces follows, e.g., from [17, Theorem 5.5.1] (for H 1/2 (∂Ω), we can use a rectification argument here).…”
Section: Wwwgamm-mitteilungenorgmentioning
confidence: 92%
“…in Ω" for u a , u b ∈ H 1 (Ω) without any danger of confusion. For more details on this topic and the involved concepts, we refer to [Bonnans, Shapiro, 2000;Christof, Müller, 2018;Harder, G. Wachsmuth, 2018].…”
Section: Notation Problem Setting and Preliminariesmentioning
confidence: 99%
“…The choice of the underlying notion of capacity is not relevant in this formulation, as the polar sets coincide on Γ C for all reasonable Sobolev capacities, cf. [8,Cor. 6.2].…”
Section: Reference Configurations Of One Body Contact Problemsmentioning
confidence: 99%
“…), if it is violated only on sets of capacity 0. With capacity theory being a complex topic itself, we refer the interested reader to an overview of Sobolev-capacity concepts and their respective behavior near the boundary of the underlying domain in [8]. In the setting at hand, it turns out that the sets of vanishing capacity, and therefore the associated notion of "q.e.…”
Section: Introductionmentioning
confidence: 99%