Almost minimizers for semilinear free boundary problems with variable coefficients

Queiroz, Olivâine S. de; Tavares, Leandro S.

doi:10.1002/mana.201600103

Cited by 12 publications

(5 citation statements)

References 19 publications

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“…However, it is not true that minimizers we are interested in are quasi-minimizers in the sense of [17], as one cannot see the Laplace-Beltrami operator as a small deformation of the Euclidian Laplacian. A similar regularity theory for div(A∇•) operators has been started in [19], though they only deal with the first step of the strategy in proving that optimal solutions are Hölder-continuous in the general case. But even if a similar regularity theory was valid for such elliptic operator, it would still remain the difficulty to prove that a solution to ( 12) is a quasi-minimizer in the sense of [17], the main difficulty here being to handle the volume constraint.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Lamboley,

Sicbaldi

2019

Preprint

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In this paper, we study the minimization of λ1(Ω), the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets Ω of fixed volume in a Riemmanian manifold (M, g). In the Euclidian setting (when (M, g) = (R n , e)), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds (M, g). In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…• prove the nondegeneracy of u near the boundary (Corollary 4. 19), so that blow-up limits are nontrivial. This requires a deeper analysis of the penalization of the volume constraint, in particular to show that the penalization parameter can be chosen positive for inner perturbations (Sections 4.4 and 4.5),…”

Section: Regularitymentioning

confidence: 99%

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Lamboley,

Sicbaldi

2019

Preprint

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“…We are interested in studying the boundary regularity of those sets of locally finite perimeter which are almost-minimizers of the F A -surface energy in an open set when compared to their local compactly contained variations. Recent work addressing regularity of almost-minimizers for other variational problems can be found in [41,16,8,26] and the notions of almost-minimizers we consider are similar. Fix universal constants n ≥ 2, 0 < λ ≤ Λ < +∞, κ ≥ 0, α ∈ (0, 1) and r 0 ∈ (0, +∞), and let A = (a ij (x)) n i,j=1 be a symmetric, uniformly elliptic, and Hölder continuous with respect to λ, Λ, and α and fix an open set U in R n .…”

Section: 2mentioning

confidence: 99%

Regularity of almost-minimizers of Hölder-coefficient surface energies

Simmons¹

2022

DCDS

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<p style='text-indent:20px;'>We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.</p>

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“…In [12] David and Toro studied properties of the free boundaries for almost-minimizers of the one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals considered in [1] and [2] respectively. They proved that, in any dimension, almost-minimizers are non-degenerate and Lipschitz continuous (see also [17]). More recently, David-Engelstein-Toro proved in [11] that, under suitable assumption, the free boundaries of almost-minimizers are uniformly rectifiable for both functionals, and almost everywhere given as the graph of a C 1,α function for the one-phase functional.…”

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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Almost minimizers for semilinear free boundary problems with variable coefficients

Cited by 12 publications

References 19 publications

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Regularity of almost-minimizers of Hölder-coefficient surface energies

Free boundary regularity for a multiphase shape optimization problem

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