Regularity for Shape Optimizers: The Nondegenerate Case

Kriventsov, Dennis; Lin, Fanghua

doi:10.1002/cpa.21743

Cited by 34 publications

(43 citation statements)

References 30 publications

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“…We prove an estimate like this in the nondegenerate case in [14]. Together, these estimates ensure that ∂Ω * is tame from a geometric measure theory perspective, and aid various blowup and compactness arguments.…”

Section: Introductionmentioning

confidence: 53%

“…The book [13] discusses some of these examples, including F which are linear, or are linear in the powers (or reciprocals), of the eigenvalues. The key point we wish to emphasize here is that unlike in our earlier work [14], here F may depend on just some subset of the eigenvalues, and not depend on some of them at all. The special case F (Ω) = λ N (Ω) is the canonical example of this, and the reader might find it useful to focus on this example for concreteness.…”

Section: Notation Vocabulary and Backgroundmentioning

confidence: 91%

“…Sketch of proof. The openness of p-minimizers can be deduced from [14][Lemma 2.5], which applies to p-minimizers. Property (2) comes from the main result in [6], for the continuity, and [14][Lemma 2.1], for the lower bound.…”

Section: Setmentioning

confidence: 99%

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Regularity for Shape Optimizers: The Degenerate Case

Kriventsov

Lin

2019

Comm Pure Appl Math

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We consider minimizers of F(λ1(Ω),…,λN(Ω))+|Ω|, where F is a function nondecreasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often‐studied F=λN. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler‐Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.

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Section: Introductionmentioning

confidence: 53%

Section: Notation Vocabulary and Backgroundmentioning

confidence: 91%

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Regularity for Shape Optimizers: The Degenerate Case

Kriventsov

Lin

2019

Comm Pure Appl Math

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“…Thus, the regularity of the optimal set Ω * is strongly related to (not to say a consequence of) the regularity of the free boundaries of the solutions of (1.1). A result for more general functionals was proved by Kriventsov and Lin [19], still under some structural assumption on the free boundary. It was then extended by the same authors to general spectral functionals in [20].…”

Section: Relation With Shape Optimization Problems For the Eigenvaluementioning

confidence: 93%

Regularity of the free boundary for the vectorial Bernoulli problem

Mazzoleni¹,

Terracini

Velichkov

2020

Analysis & PDE

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In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D ⊂ R d , Λ > 0 andWe prove that, for any optimal vector U = (u 1 , . . . , u k ), the free boundary ∂(∪ k i=1 {u i = 0}) ∩ D is made of a regular part, which is relatively open and locally the graph of a C ∞ function, a (one-phase) singular part, of Hausdorff dimension at most d − d * , for a d * ∈ {5, 6, 7}, and by a set of branching (two-phase) points, which is relatively closed and of finite H d−1 measure. Our arguments are based on the NTA structure of the regular part of the free boundary.

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“…, n, is C 1,α regular (Theorem 1.8). This line of study has become increasingly important in recent years, where regularity results for solutions of free-boundary problems, and in particular almost-minimizers, have been applied to study the regularity of shape optimization problems involving eigenvalues of the Dirichlet-Laplacian (see for instance [16,14,15,9]).…”

Section: Introductionmentioning

confidence: 99%

Free boundary regularity for a multiphase shape optimization problem

Spolaor

Trey

Velichkov

2019

Communications in Partial Differential Equations

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In this paper we prove a C 1,α regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a new application of the epiperimetric-inequality of [18] up to the boundary. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension. 1 Remark 1.2. The Hölder continuity of the (exterior) normal vector n Ω is the best regularity result that one can expect. Indeed, recently Chang-Lara and Savin [10] showed that even for minimizers the regularity of the constrained free boundaries cannot exceed C 1, 1 /2 . Moreover, we notice that the result analogous to Theorem 1.1 was proved in any dimension in [10], by a viscosity approach, but only for minimizers of the functional J op .Analogously, we say that the nonnegative function u : B 2 → R is a almost-minimizer of the one-phase functional J op in B 2 , if u ∈ H 1 (B 2 ) and there are constants r 1 > 0, C 1 > 0 and δ 1 > 0 such that, for every x 0 ∈ B 1 ∩ ∂Ω u and r ∈ (0, r 1 ), we haveThe regularity of the unconstrained one-phase free boundary ∂Ω u follows directly by Theorem 1.1. For the sake of completeness, we give the precise statement in Corollary 1.3 below.Corollary 1.3 (Regularity of the unconstrained one-phase free boundaries). Let B 2 ⊂ R 2 and u : B 2 → R be a non-negative and Lipschitz continuous function. If u is a almost-minimizer of the functional J op in B 2 , then the free boundary B 1 ∩ ∂Ω u is locally the graph of a C 1,α function.

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Regularity for Shape Optimizers: The Nondegenerate Case

Cited by 34 publications

References 30 publications

Regularity for Shape Optimizers: The Degenerate Case

Regularity for Shape Optimizers: The Degenerate Case

Regularity of the free boundary for the vectorial Bernoulli problem

Free boundary regularity for a multiphase shape optimization problem

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