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

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