Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

TL;DR

This paper establishes geometric inequalities for the lowest Robin Laplacian eigenvalues on 2-manifolds with boundary and extends the results to unbounded 3D cones, identifying optimal shapes under perimeter constraints.

Contribution

It proves that geodesic disks maximize the lowest Robin eigenvalue among manifolds with bounded curvature and fixed perimeter, and extends spectral isoperimetric inequalities to unbounded cones.

Findings

Geodesic disks maximize the Robin eigenvalue under curvature and perimeter constraints.

Spectral isoperimetric inequality holds for the Dirichlet-to-Neumann operator.

Circular cones maximize the Robin eigenvalue among unbounded cones with fixed cross-section perimeter.

Abstract

We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_\circ \ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_\circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.