TY - JOUR
T1 - Spectral optimization for the Stekloff-Laplacian
T2 - The stability issue
AU - Brasco, Lorenzo
AU - De Philippis, Guido
AU - Ruffini, Berardo
N1 - Funding Information:
The authors wish to warmly thank Giuseppe Buttazzo for some discussions on the topic of this paper, as well as for having stimulated the writing of it. We also thank Alessio Figalli for having kindly suggested to us a simple proof of Lemma 6.2. An anonymous referee is gratefully acknowledged for the careful reading of the manuscript and for the many useful comments. L.B. has been partially supported by the ERC Advanced Grant No. 226234, while G.D.P. acknowledges the support of the ERC Advanced Grant No. 246923.
PY - 2012/6/1
Y1 - 2012/6/1
N2 - 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.
AB - 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.
KW - Stability for eigenvalues
KW - Stekloff boundary value problem
KW - Weighted isoperimetric inequality
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U2 - 10.1016/j.jfa.2012.03.017
DO - 10.1016/j.jfa.2012.03.017
M3 - Article
AN - SCOPUS:84859434987
SN - 0022-1236
VL - 262
SP - 4675
EP - 4710
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 11
ER -