Minimizing eigenvalues of the fractional Laplacian

We study the minimizers of λ ₖˢ(A) + |A| λ k s ( A ) + | A | where λ 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 λ 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.