Area minimizing sets subject to a volume constraint in a convex set

Stredulinsky, Edward W.; Ziemer, William P.

doi:10.1007/bf02921639

Cited by 58 publications

(83 citation statements)

References 9 publications

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“…Since the sets C s are convex minimizers of P(E) − µ s |E| among all E ⊆ C, for s ≥ s * , their boundary is of class C 1,1 [Brézis and Kinderlehrer 1974;Stredulinsky and Ziemer 1997], with curvature at most µ s , and equal to µ s in the interior of C (note that µ s = h C for s ∈ [s * , s * ]). Remark 3.2.…”

Section: Proof Let Smentioning

confidence: 99%

“…We may assume that near x, ∂C is the graph of a nonnegative, C 2 and convex function f : B → ‫ޒ‬ where B is an (N − 1)-dimensional ball centered at x. We may as well assume that ∂C s is the graph of f s : B → ‫,ޒ‬ which is C 1,1 [Brézis and Kinderlehrer 1974;Stredulinsky and Ziemer 1997], and also nonnegative and convex. In B, we have f s ≥ f ≥ 0, and…”

Section: Proof Let Smentioning

confidence: 99%

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Uniqueness of the Cheeger set of a convex body

Caselles¹,

Chambolle²,

Novaga³

2007

Pacific J. Math.

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Section: Proof Let Smentioning

confidence: 99%

Section: Proof Let Smentioning

confidence: 99%

Uniqueness of the Cheeger set of a convex body

Caselles¹,

Chambolle²,

Novaga³

2007

Pacific J. Math.

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“…The existence of isoperimetric regions for any given volume is solved in the context of sets of finite perimeter, see [6,Chapter 1]. Regularity questions have been studied by Gonzalez, Massari and Tamanini [7] and by Stredulinsky and Ziemer [16]. They have proved that a minimizer E satisfies that ∂E ∩ Ω is a smooth hypersurface with constant mean curvature H 0 off of a singular closed set of small Hausdorff dimension.…”

Section: P (E) ≤ P (E )mentioning

confidence: 99%

Isoperimetric regions in rotationally symmetric convex bodies

Rosales¹

2003

Indiana Univ. Math. J.

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“…∂ Ω is of class C 1,1 , then also ∂C enjoys the same regularity (see [4]); the same result holds if Ω is convex, as a consequence of the results in [21].…”

Section: Definitions and Preliminary Resultsmentioning

confidence: 82%

Shape derivative of the Cheeger constant

Parini

Saintier

2014

ESAIM: COCV

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ABSTRACT. This paper deals with the existence of the shape derivative of the Cheeger constant h 1 (Ω) of a bounded domain Ω. We prove that if Ω admits a unique Cheeger set, then the shape derivative of h 1 (Ω) exists, and we provide an explicit formula. A counterexample shows that the shape derivative may not exist without the uniqueness assumption. INTRODUCTIONLet Ω ⊂ R n be a bounded domain. The Cheeger constant of Ω is defined asHere P(E; R n ) is the distributional perimeter of E measured with respect to R n , while |E| is the n−dimensional Lebesgue measure of E. A set C ⊂ Ω for which the infimum is attained is called a Cheeger set.The problem of finding a Cheeger set for a given domain Ω has extensively received attention in the last decades, starting from the original work of Jeff Cheeger [5]. For an introductory survey on the Cheeger problem we refer to [18]; here we recall that for every bounded domain Ω with Lipschitz boundary there exists at least one Cheeger set. Uniqueness does not hold in general, but it is guaranteed if we assume Ω to be convex; in this case the Cheeger set turns out to be convex and of class C 1,1 (see [1]). The Cheeger constant can be obtained as the limit for p → 1 of the first eigenvalue λ p (Ω) of the p−Laplacian under Dirichlet boundary conditions (see [12]), and corresponds to the first eigenvalue of the 1−Laplacian (see [14]).Shape analysis roughly consists in studying the regularity and the optimisation of a functional J : Ω ∈ A → J(Ω) ∈ R defined over some class A of subsets Ω ⊂ R n . Due to its physical relevance, a particularly important class of functionals are the ones defined in terms of the eigenvalues of some operator. A lot of works have been dedicated for instance to the study of the dependence of the eigenvalues of the Laplacian as functions of the domain under various boundary conditions. We refer for example to the monograph [11] for an introduction to the field of shape analysis.In order to optimize J over A it is important to determine how sensitive is J under perturbation of a given set Ω. Given a smooth vector field V ∈ C ∞ c (R n ; R n ), define F t : R n → R n as F t (x) = (Id + tV )(x). We then perturb Ω in the direction V by considering the sets Ω t = F t (Ω). The shape derivative of J in the direction V at Ω is then defined as J(Ω,V ) ′ := lim t→0 J(Ω t ) − J(Ω) t .

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Area minimizing sets subject to a volume constraint in a convex set

Cited by 58 publications

References 9 publications

Uniqueness of the Cheeger set of a convex body

Uniqueness of the Cheeger set of a convex body

Isoperimetric regions in rotationally symmetric convex bodies

Shape derivative of the Cheeger constant

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