Inserted: 22 nov 2011
Last Updated: 6 feb 2013
Journal: J. Funct. Anal.
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.
Keywords: Stability for eigenvalues, Weighted isoperimetric inequality