Symmetry-breaking and local stability of a two-phase eigenvalue problem in optimal insulation

We consider the first eigenvalue, $\lambda_\beta(\Omega,A)$, of a two-phase eigenvalue problem for the Laplacian with Robin boundary conditions, where the two phases are characterised by different ellipticity constants. We characterise the conditions under which a ball $B_R$ is a local minimum under a volume constraint for the minimisation problem $A\mapsto\lambda_\beta(B_r,A)$, in terms of the principal Neumann eigenvalue of the fixed inner ball $B_r$.