Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

Egert, Moritz; Haller-Dintelmann, Robert; Rehberg, Joachim

doi:10.1007/s11118-015-9463-8

Cited by 28 publications

(25 citation statements)

References 41 publications

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“…Usually we encounter the spaces W 1,p D k (Ω). Under Assumption N there are bounded linear Sobolev extension operators E k that extend [30,Thm. 6.16].…”

Section: Sobolev Spacesmentioning

confidence: 99%

L -estimates for the square root of elliptic systems with mixed boundary conditions

Egert

2018

Journal of Differential Equations

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This article focuses on L p -estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions on domains beyond the Lipschitz class. If there is an associated bounded semigroup on L p 0 , then we prove that the square root extends for all p ∈ (p0, 2) to an isomorphism between a closed subspace of W 1,p carrying the boundary conditions and L p . This result is sharp and extrapolates to exponents slightly above 2. As a byproduct, we obtain an optimal p-interval for the bounded H ∞ -calculus on L p . Estimates depend holomorphically on the coefficients, thereby making them applicable to questions of (non-autonomous) maximal regularity and optimal control. For completeness we also provide a short summary on the Kato square root problem in L 2 for systems with lower order terms in our setting.

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“…Usually we encounter the spaces W 1,p D k (Ω). Under Assumption N there are bounded linear Sobolev extension operators E k that extend [30,Thm. 6.16].…”

Section: Sobolev Spacesmentioning

confidence: 99%

L -estimates for the square root of elliptic systems with mixed boundary conditions

Egert

2018

Journal of Differential Equations

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“…Let us point out that sets satisfying the lower density estimate (4.2), are sometimes also called (d−1)thick, see e.g. [Leh08,EHDR15]. The proof of Proposition 4.4 is carried out by contradiction to (4.1).…”

Section: Limit Passage In the Semistability Condition And Fine Propertimentioning

confidence: 99%

From adhesive to brittle delamination in visco-elastodynamics

Rossi

Thomas

2017

Math. Models Methods Appl. Sci.

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In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.2010 Mathematics Subject Classification: 49J53, 49J45, 74H20, 74C05, 74C10, 74M15, 74R10.The asymptotic analysis as k → ∞ for the purely rate-independent adhesive contact system, coupling the flow rule (1.3d) (with no regularizing gradient term), with the static momentum balance, was carried out in [RSZ09] by resorting to the evolutionary Γ-convergence results for rate-independent processes from [MRS08]. Loosely speaking, the main observation is that the adhesive contact contribution ΓC k 2 z|[[u]]| 2 dH d−1 to E k (1.6) penalizes displacement jumps in points with positive z, and leads as k → ∞], z) = ∞ otherwise, and G ∞ the Γ-limit as k → ∞ of the perimeter energies (G k ) k from (1.4). Let us stress that, adapting some arguments from [DML11], we shall prove that along semistable energetic solutions of the brittle system, the energy-dissipation inequality actually holds as a balance, along any arbitrary interval [s, t] ⊂ [0, T ] for almost all s ≤ t ∈ (0, T ), and for s = 0.Let us finally mention that our ansatz for G k and R k , cf. (1.4) and (1.7), will allow for different scalings of the parameters a 0 k , a 1 k , and b k , cf. (2.15). In this way, we can obtain different fracture models in the brittle limit. We will discuss the different options in Section 2.3.Plan of the paper. In Section 2 we give our weak solvability notion for damped inertial systems with a mixed rate-independent/rate-dependent character. In particular, in Sec. 2.1 we specify it in the context of the adhesive contact model and then state the existence of semistable energetic solutions to the adhesive system. In Sec. 2.2 we give the notion of semistable energetic solutions to the brittle system, while in Sec. 2.3 we present our main result, Theorem 2.10, which provides the existence of semistable energetic solutions for the brittle model in terms of an approximation result via the adhesive contact systems. We also compare our result with other existing results on dynamic fracture.The existence...

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“…We exclude this case here to avoid discussing the problems and peculiarities that arise in the context of Hardy's inequality and the inverse trace theorem when W 1,1 -spaces are considered, cf. [22] and [23].…”

Section: Capacity Theory and Sobolev Spacesmentioning

confidence: 99%

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

Christof

Müller

2018

GAMM-Mitteilungen

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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.

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Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

Abstract: Abstract. We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.

Cited by 28 publications

References 41 publications

L -estimates for the square root of elliptic systems with mixed boundary conditions

L -estimates for the square root of elliptic systems with mixed boundary conditions

From adhesive to brittle delamination in visco-elastodynamics

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

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