# Robin spectrum: two disks maximize the third eigenvalue

## TL;DR

This paper establishes that among simply-connected planar domains with fixed area, the third Robin Laplacian eigenvalue is maximized by a union of two disks under certain Robin parameter conditions, with equality in degenerate cases.

## Contribution

It proves a sharp upper bound for the third Robin eigenvalue using a union of two disks, extending eigenvalue optimization results to Robin boundary conditions.

## Key findings

- The third Robin eigenvalue is maximized by two disks for certain parameters.
- Equality occurs when the domain degenerates into two disks.
- The result generalizes known bounds for Dirichlet and Neumann cases.

## Abstract

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.