Minimizing eigenvalues of the fractional Laplacian

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.