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arXiv · 1909.10842

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

Magda Khalile·Vladimir Lotoreichik
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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.

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Magda Khalile, Vladimir Lotoreichik. 2019-11-13. Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones. https://arxiv.org/abs/1909.10842

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