Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues

Details Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues

2009-06-05

arxiv.org/abs/0906.1091v1

Mathematics

Analysis of PDEs

17 pages. To appear in JEMS

Scientific paper

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.

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