Minimizing eigenvalues of the fractional Laplacian

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2026年2月17日

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Alvis Zahl

摘要

We study the minimizers of $\lambda _k^s(A) + |A|$ $λ_{k}^{s} (A) + | A |$ where $\lambda ^s_k(A)$ $λ_{k}^{s} (A)$ 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 $\lambda _k$ $λ_{k}$ 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.