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 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.

On the Non‐Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces

“…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).…”

On Second-Order Optimality Conditions for Optimal Control Problems Governed by the Obstacle Problem

“…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].…”

On the control of time discretized dynamic contact problems

“…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].…”

“…), 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.…”