Minimizing Eigenvalues of the Fractional Laplacian

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.