On reverse Faber-Krahn inequalities

In 1961, Payne and Weinberger showed that ‘among the class of membranes with given area, free along the interior boundaries and fixed along the outer boundary of given length, the concentric annulus has the highest fundamental frequency.’ In the present study, we extend this result for the first eigenvalue of the p-Laplacian, for p∈(1,∞), in higher dimensional domains where the outer boundary is a fixed sphere. As an application, we prove that the nodal set of the second eigenfunctions of the p-Laplacian (with mixed boundary conditions) on a ball cannot be a concentric sphere.