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The research aims to investigate minimizers of the k-th Dirichlet eigenvalue of the fractional Laplacian and their properties.

February 19, 2026Open Access

Minimizing eigenvalues of the fractional Laplacian

Puntos clave

The research aims to investigate minimizers of the k-th Dirichlet eigenvalue of the fractional Laplacian and their properties.
Analysis of the fractional Laplacian's Dirichlet eigenvalue minimizers.
Exploration of optimal Hölder regularity for eigenfunctions.
Development of a combinatorial problem connected to global configurations of minimizers.
Demonstrated existence of minimizers for the fractional Laplacian.
Characterized optimal regularity of associated eigenfunctions.
Identified distinct global behaviors of free boundaries for minimizers.

Resumen

Abstract We study the minimizers of ₖˢ (A) + |A| λ k s (A) + | A | where ˢₖ (A) λ k s (A) is the k -th Dirichlet eigenvalue of the fractional Laplacian on A. Unlike in the case of the Laplacian, free boundary of minimizers exhibits distinct global behaviors. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where ₖ 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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