On the Hong-Krahn-Szego inequality for the p-Laplace operator

Given an open set ?, we consider the problem of providing sharp lower bounds for ? (?), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong-Krahn-Szego inequality, asserting that the disjoint unions of two equal balls minimize ? among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ? are considered as well. © 2012 Springer-Verlag Berlin Heidelberg.