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

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