Laplacian eigenvalues for mean zero functions with constant Dirichlet data

We investigate the eigenvalues of the Laplace operator in the space of functions of mean zero and having a constant (unprescribed) boundary value. The first eigenvalue of such problem lies between the first two eigenvalues of the Laplacian with homogeneous Dirichlet boundary conditions and satisfies an isoperimetric inequality: in the class of open bounded sets of prescribed measure, it becomes minimal for the union of two disjoint balls having the same radius. Existence of an optimal domain in the class of convex sets is also discussed.