Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk

TL;DR

This paper investigates the extremal properties of the first and second mixed eigenvalues of the Laplacian on a disk with Dirichlet-Neumann boundary conditions, identifying explicit minimizers and maximizers within certain boundary partitions.

Contribution

It provides explicit descriptions of the boundary conditions that minimize the second eigenvalue and maximize the first eigenvalue among boundary partitions, advancing understanding of spectral optimization.

Findings

The second eigenvalue's minimizer forms a specific 1-parameter family with an explicit description.

The first eigenvalue is maximized by a uniformly distributed boundary partition.

Explicit characterization of extremal boundary conditions for mixed eigenvalues.

Abstract

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.