Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign
Catherine Bandle, Alfred Wagner · St Petersburg Mathematical Journal · 2017
The energy of the torsion problem with Robin boundary conditions is considered in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive elasticity constants, the ball does not provide a minimum. For nearly spherical domains and elasticity constants close to zero, the energy is the largest for the ball. This result is true for general domains in the plane under an additional condition on the first nontrivial Steklov eigenvalue. For more negative elasticity constants the situation is more involved and is strongly related to the particular domain perturbation. The methods used in the paper are the series representation of the solution in terms of Steklov eigenfunctions, the first and second shape derivatives, and an isoperimetric inequality of Payne and Weinberger for the torsional rigidity.