The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings W 1,2 0 (Ω) → L q (Ω).2010 Mathematics Subject Classification. 47A75, 49Q20, 49R05.
In this paper we study the regularity of the optimal sets for the shape optimization problemwhere λ 1 (·), . . . , λ k (·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ω * k is composed of a relatively open regular part which is locally a graph of a C ∞ function and a closed singular part, which is empty if d < d * , contains at most a finite number of isolated points if d = d * and has Hausdorff dimension smaller than (d − d * ) if d > d * , where the natural number d * ∈ [5, 7] is the smallest dimension at which minimizing one-phase free boundaries admit singularities.To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.
This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min{g ((F 1 For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., F i = λ k i .
In this paper we prove that the shape optimization problemwhere λ k (Ω) is the kth eigenvalue of the Dirichlet Laplacian, has a solution for any k ∈ N and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d − 8. Our results apply to general spectral functionals of the form f (λ k1 (Ω), . . . , λ kp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.
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