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• ≥ 0 and under the constraint of fixed perimeter, the geodesic disk of constant curvature K• 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 threedimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.

Optimization of the lowest eigenvalue of a soft quantum ring

“…In the course of the analysis we employ the co-area formula to prove that under transplantation the kinetic energy R 2 |∇u| 2 dx and the L 2 -norm of the function do not increase, and we use the total curvature identity to show that the potential energy R 2 |u| 2 dµ is preserved. This strategy of the proof is simple but powerful; it was recently successfully applied to the Robin Laplacian on bounded domains [AFK17, BFNT18, FK15], on 2manifolds [KL19], and on exterior domains [KL18,KL20], and also to the two-dimensional Schrödinger operator with δ ′ -interaction supported on a closed curve [L18].…”

Asymptotics of Robin eigenvalues on sharp infinite cones

“…By the estimate (6) in Lemma 3 we can control the term that contains ∂ s ψ. Namely, for s ∈ (0, a) one has ρ(s, ε) ∈ (0, a), and then one can find some K > 0 such that for any s ∈ (0, a) there holds…”

“…For n = 2, the accumulation of eigenvalues at −1 was studied in greater detail in [1]: for δ → 0 + the number N δ of discrete eigenvalues in (−∞, −1 − δ) behaves as N δ ∼ 1 4δ cos 2 θ sin θ . Furthermore, in [6] it was shown that Q ε maximizes the first eigenvalue among all cones with the same perimeter of the spherical cross-section.…”

On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter

“…Spectral optimization for the lowest Robin eigenvalue in other geometric settings is considered in e.g. [26,30,31]. Isoperimetric inequalities for higher Robin eigenvalues are obtained in [17,18,21].…”