Eigenvalue optimization via a first-variation formula

Abstract

We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum. This formula provides an effective tool for describing the structure of critical points in eigenvalue optimization problems and can also yield simple proofs of the existence of optimizers. We illustrate these advantages through applications to the optimization of weighted Laplace and Steklov eigenvalues. In particular, we characterize all optimal weights, thereby answering some open questions posed by Kokarev, and give a short proof that such weights exist.

28 pages; finite-dimensionality results have been added for the closed-manifold case of Theorem 1.1; see also Remark 1.3