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

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 ı 0 and under the constraint of fixed perimeter, the geodesic disk of constant curvature K ı 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.

On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

“…In the context of Riemannian manifolds, Khalile and Lotoreichik in [24] proved the following. Let Ω be a compact, two-dimensional, simply connected Riemannian manifold with C 2 boundary and with Gauss curvature bounded from above by a non-negative constant κ 0 , and let B be a geodesic disc in the simply connected space form of Gauss curvature κ 0 with the same perimeter as Ω.…”