Second variation of domain functionals and applications to problems with Robin boundary conditions

Catherine Bandle, Alfred Wagner

Published 2014-03-10, updated 2015-07-10Version 2

In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular domains of given volume are derived. The discussion leads to the investigation of the eigenvalues of a Steklov eigenvalue problem. As a byproduct a general characterization of the optimal shapes is obtained.