Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues

Abstract

This paper is devoted to the study of Lyapunov-type inequalities for Neumann boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis of the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the use of suitable minimization problems. This method of proof yields new information on Lyapunov constants. For instance, we prove that as in the classical result by Lyapunov, the best constant is not attained. Additionally, we exploit the relation between Neumann boundary conditions and disfocality to provide new nonresonance conditions at higher eigenvalues. © European Mathematical Society 2010.