Minimizing Eigenvalues of the Fractional Laplacian

TL;DR

This work analyzes the optimization problem of minimizing the $k$-th Dirichlet eigenvalue of the fractional Laplacian plus the volume, $\lambda_k^s(A) + |A|$, over admissible sets. The authors reformulate the problem in a union-of-copies framework to handle nonlocal interactions, establish the existence and boundedness of minimizers, and prove optimal $C^{0,s}$ regularity for eigenfunctions. In the simple eigenvalue case, they obtain non-degeneracy, blow-up analysis with Weiss monotonicity, and separation/density results for the free boundary, culminating in $C^{1,\gamma_0}$ regularity of the free boundary via an almost-minimizer framework. Finally, they discuss the global configuration and a discrete toy model to capture interactions between disconnected minimizer components, including conjectures about the structure of minimizers and the limiting behavior as $s \to 1$.

Abstract

We study the minimizers of \begin{equation} λ_k^s(A) + |A| \end{equation} where $λ^s_k(A)$ 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 Hölder regularity for the corresponding eigenfunctions, and in the case where $λ_k$ 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.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.6
• Proposition 2.7
• proof