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arXiv · 2504.09840

Minimizing Eigenvalues of the Fractional Laplacian

Alvis Zahl
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Abstract

We study the minimizers of \begin{equation} \lambda_k^s(A) + |A| \end{equation} where $\lambda^s_k(A)$ 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 H\"older regularity for the corresponding eigenfunctions, and in the case where $\lambda_k$ 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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Alvis Zahl. 2025-04-14. Minimizing Eigenvalues of the Fractional Laplacian. https://doi.org/10.1007/s00526-026-03270-z

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