Spectral optimization for the Stekloff–Laplacian: The stability issue

We consider the problem of minimizing 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. (1999) 2]: we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.