An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

0566557 - ÚJF 2024 RIV US eng J - Journal Article
Lotoreichik, Vladimir
An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain.
Journal of Differential Equations. Roč. 345, FEB (2023), s. 285-313. ISSN 0022-0396. E-ISSN 1090-2732
R&D Projects: GA ČR(CZ) GA21-07129S
Institutional support: RVO:61389005
Keywords : Bi-Laplacian * Robin boundary condition * Planar exterior domain * Isoperimetric inequality * Parallel coordinates * min-max principle
OECD category: Pure mathematics
Impact factor: 2.4, year: 2022
Method of publishing: Limited access
https://doi.org/10.1016/j.jde.2022.11.016

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C2-smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold.
Permanent Link: https://hdl.handle.net/11104/0337876