Existence and Regularity Results for Some Shape Optimization Problems

“…A natural further question is of course whether the convexity assumption can be dropped, and instead consider the optimization problem (3) in the class of quasi-open sets. As mentioned in the introduction the existence of optimizers for this problem with γ ≥ 1 is covered by the results in [9,13,45,46] (see also [44,55]). The results in these articles consider the case of functions of a fixed number of eigenvalues which is not the case for (2).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

Larson

2019

J. Spectr. Theory

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For Ω ⊂ R n , a convex and bounded domain, we study the spectrum of −∆Ω the Dirichlet Laplacian on Ω. For Λ ≥ 0 and γ ≥ 0 let ΩΛ,γ (A) denote any extremal set of the shape optimization problem, where A is an admissible family of convex domains in R n . If γ ≥ 1 and {Λj } j≥1 is a positive sequence tending to infinity we prove that {ΩΛ j ,γ (A)} j≥1 is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then ΩΛ,γ converges, up to rotation and translation, to the regular m-gon.2010 Mathematics Subject Classification. 35P15, 47A75, 49Q10.

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“…The rest of the proof follows precisely as in [39,Lemma 3.51]. For every 0 < t < u L ∞ and ε > 0, we consider the test function…”

Section: 1mentioning

confidence: 99%

Existence and regularity of optimal shapes for elliptic operators with drift

2019

Self Cite

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This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = −∆+V (x)·∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(Ω, V ) for a bounded quasi-open set Ω which enjoys similar properties to the case of open sets. Then, given m > 0 and τ ≥ 0, we show that the minimum of the following non-variational problemThe existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved.The second interest and main result of this paper is the regularity of the optimal shape Ω * solving the minimization problemwhere Φ is a given Lipschitz function on D. We prove that the optimal set Ω * is open and that its topological boundary ∂Ω * is composed of a regular part, which is locally the graph of a C 1,α function, and a singular part, which is empty if d < d * , discrete if d = d * and of locally finite, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x ∈ ∂Ω * ∩ ∂D, ∂Ω * is C 1,1/2 in a neighborhood of x.

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Existence and Regularity Results for Some Shape Optimization Problems

Cited by 23 publications

References 72 publications

A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

Existence and regularity of optimal shapes for elliptic operators with drift

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