Lipschitz Regularity of the Eigenfunctions on Optimal Domains

Bucur, Dorin; Mazzoleni, Dario; Pratelli, Aldo; Velichkov, Bozhidar

doi:10.1007/s00205-014-0801-6

Cited by 34 publications

(60 citation statements)

References 21 publications

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“…, k. Contrary to the subsolutions which were used to get information on the optimal shape, the supersolutions will be useful to gather information on the state functions (here, the eigenfunctions of the optimal shape). The main tool is the following notion of quasi-minimality [9,16].…”

Section: Shape Supersolutions and Quasi-minimizersmentioning

confidence: 99%

“…Following the ideas of Briançon et al [6], it was proved in [9,12] that the quasi-minimizers which are eigenfunctions of their own support are globally Lipschitz. Indeed, we recall the following result [9,12,16].…”

Section: Definition 32 We Say That a Function U ∈ H 1 (R D ) Is Quamentioning

confidence: 99%

“…In order to prove that the result above can apply to minimizers of (1.2), we restate a lemma which was proved in [9] and will allow us to handle the Rayleigh quotient for the perturbed domains. We shall analyse first the simpler problem (1.3), and then show how to use the supersolution property to pass from (1.3) to the general case (1.2).…”

Section: Then There Is a Constant M Depending Only On λ And C Such Tmentioning

confidence: 99%

“…As the Lipschitz constant is independent on , we can pass to the limit as → 0 obtaining on the limit an eigenfunction on Ω * which is still globally Lipschitz (for the details on the approximation argument, we refer to [9]). …”

Section: Then There Is a Constant R > 0 Such That For Every Mapmentioning

confidence: 99%

“…Consequently, Ω * has a Lipschitz eigenfunction corresponding to λ i j (Ω). For more details and precise results, we refer to [9].…”

Section: Then There Is a Constant R > 0 Such That For Every Mapmentioning

confidence: 99%

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A free boundary approach to shape optimization problems

Bucur

Velichkov

2015

Phil. Trans. R. Soc. A.

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The analysis of shape optimization problems involving the spectrum of the Laplace operator, such as isoperimetric inequalities, has known in recent years a series of interesting developments essentially as a consequence of the infusion of free boundary techniques. The main focus of this paper is to show how the analysis of a general shape optimization problem of spectral type can be reduced to the analysis of particular free boundary problems. In this survey article, we give an overview of some very recent technical tools, the so-called shape suband supersolutions, and show how to use them for the minimization of spectral functionals involving the eigenvalues of the Dirichlet Laplacian, under a volume constraint.

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Section: Shape Supersolutions and Quasi-minimizersmentioning

confidence: 99%

Section: Definition 32 We Say That a Function U ∈ H 1 (R D ) Is Quamentioning

confidence: 99%

Section: Then There Is a Constant M Depending Only On λ And C Such Tmentioning

confidence: 99%

Section: Then There Is a Constant R > 0 Such That For Every Mapmentioning

confidence: 99%

“…Consequently, Ω * has a Lipschitz eigenfunction corresponding to λ i j (Ω). For more details and precise results, we refer to [9].…”

Section: Then There Is a Constant R > 0 Such That For Every Mapmentioning

confidence: 99%

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A free boundary approach to shape optimization problems

Bucur

Velichkov

2015

Phil. Trans. R. Soc. A.

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Regularity for Shape Optimizers: The Nondegenerate Case

Kriventsov

Lin

2018

Comm Pure Appl Math

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We consider minimizers of F(λ1(Ω),…,λN(Ω))+| Ω |, where F is a function strictly increasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.

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An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case

Spolaor

Velichkov

2018

Comm Pure Appl Math

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Using a direct approach, we prove a two‐dimensional epiperimetric inequality for the one‐phase problem in the scalar and vectorial cases and for the double‐phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one‐phase and double‐phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc.

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Lipschitz Regularity of the Eigenfunctions on Optimal Domains

Abstract: We study the optimal sets Ω

Cited by 34 publications

References 21 publications

A free boundary approach to shape optimization problems

A free boundary approach to shape optimization problems

Regularity for Shape Optimizers: The Nondegenerate Case

An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case

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