We consider the problem of maximizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock--Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (J. of Inequal. \& Appl. 4: 215--240, 1999): we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.
Spectral optimization for the Stekloff-Laplacian: the stability issue
BRASCO, Lorenzo;
2012
Abstract
We consider the problem of maximizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock--Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (J. of Inequal. \& Appl. 4: 215--240, 1999): we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.File in questo prodotto:
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