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Pré-Publication, Document De Travail Année : 2020

Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

Résumé

In this paper we consider minimizers of the functional min λ1(Ω) + · · · + λ k (Ω) + Λ|Ω|, : Ω ⊂ D open where D ⊂ R d is a bounded open set and where 0 < λ1(Ω) ≤ · · · ≤ λ k (Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets Ω * have finite perimeter and that their free boundary ∂Ω * ∩ D is composed of a regular part, which is locally the graph of a C 1,α-regular function, and a singular part, which is empty if d < d * , discrete if d = d * and of Hausdorff dimension at most d − d * if d > d * , for some d * ∈ {5, 6, 7}.
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Dates et versions

hal-02526934 , version 1 (31-03-2020)

Identifiants

  • HAL Id : hal-02526934 , version 1

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Baptiste Trey. Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form. 2020. ⟨hal-02526934⟩
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