Minimizing Eigenvalues of the Fractional Laplacian

2025/04/14 by Alvis Zahl, Zahl, Alvis · 1 voice · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Nonlinear Partial Differential Equations #math.AP #msc:35B65 #msc:35R35

paper · pdf · doi:10.48550/arxiv.2504.09840

published as Calc. Var. 65, 98 (2026)

openalex publication_date 2025/04/14 · openalex created_date 2025/10/14 · arxiv created 2025/11/24 · openalex updated_date 2026/07/28 · arxiv updated 2026/07/30

Abstract

We study the minimizers of λ_k^s(A) + |A| where λ^s_k(A) is the k-th Dirichlet eigenvalue of the fractional Laplacian on A. Unlike in the case of the Laplacian, the free boundary of minimizers exhibit distinct global behavior. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where λ_k is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers.

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• Vectorial Bernoulli Problems and Free Boundary Systems

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