Minimizing Eigenvalues of the Fractional Laplacian

Abstract

We study the minimizers of \begin{equation} λ_k^s(A) + |A| \end{equation} where ${\lambda }_k^s(A)$ 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 ${\lambda }_k$ 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.

References in corpus (6)

• An extension problem related to the fractional Laplacian
• Nonlocal equations with measure data
• A minimization problem with free boundary related to a cooperative system
• Lipschitz regularity of the eigenfunctions on optimal domains
• Integro-Differential Elliptic Equations
• Regularity in a one-phase free boundary problem for the fractional Laplacian