Minimizing eigenvalues of the fractional Laplacian
Alvis Zahl · Calculus of Variations and Partial Differential Equations · 2026
Abstract 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.