Quantitative Spectral Stability for the Robin Laplacian

TL;DR

This work establishes quantitative spectral stability for the Robin Laplacian under the singular perturbation of removing a shrinking hole from a bounded domain. It combines variational perturbation theory, torsional rigidity analysis, and a blow-up method to derive sharp asymptotics for simple Robin eigenvalues, showing that the rate depends on whether the limit eigenfunction vanishes at the hole center; in the non-vanishing case the rate is of order \\varepsilon^{N-1} with sign matching the Robin parameter, while in the nodal case the rate is governed by the vanishing order and a torsion-like limit problem in higher dimensions. For round holes, the authors obtain explicit coefficient formulas in all dimensions, relating the expansion to the derivatives of the eigenfunction at the concentration point. In dimension two, a general blow-up argument is adapted to yield estimates, with fully explicit results available for round holes. The results deepen the understanding of how Robin eigenelements react to domain perturbations and have potential applications in shape optimization and inverse problems.

Abstract

This paper deals with eigenelements of the Laplacian in bounded domains, under Robin boundary conditions, without any assumption on the sign of the Robin parameter. We quantify the asymptotics of the variation of simple eigenvalues under the singular perturbation produced by removing a shrinking set and imposing the same Robin condition on its boundary. We also study the convergence rate of the corresponding eigenfunctions.

Theorems & Definitions (55)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9
• Proposition 3.1