Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues

TL;DR

This paper investigates Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues, providing new bounds and insights into the distribution of solutions and their derivatives, with implications for nonresonance conditions.

Contribution

It introduces a novel analysis of zeros of solutions and derivatives, deriving new Lyapunov constants and nonresonance conditions at higher eigenvalues.

Findings

Best Lyapunov constant is not attained.

New nonresonance conditions at higher eigenvalues.

Enhanced understanding of zeros distribution of solutions.

Abstract

This paper is devoted to the study of Lyapunov-type inequality for Neumann boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis about 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 allows to obtain 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.