Regularity of the optimal sets for the second Dirichlet eigenvalue

2020/10/01 by Mazzoleni, Dario, Trey, Baptiste, Velichkov, Bozhidar · 2 citations
#35R35 #47A75 #49Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2010.00441

Abstract

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set Ω minimizes the functional \mathcal F_Λ(Ω)=λ₂(Ω)+Λ|Ω|, among all subsets of a smooth bounded open set D⊂ ℝ^d, where λ₂(Ω) is the second eigenvalue of the Dirichlet Laplacian on Ω and Λ>0 is a fixed constant, then Ω is equivalent to the union of two disjoint open sets Ω₊ and Ω_-, which are C^1,α-regular up to a (possibly empty) closed set of Hausdorff dimension at most d-5, contained in the one-phase free boundaries D∩ ∂Ω₊∖∂Ω_- and D∩∂Ω_-∖∂Ω₊.

Cited by

• Vectorial Bernoulli Problems and Free Boundary Systems
• Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

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