# Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

@article{Brandolini2013OptimalLB, title={Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems}, author={B. Brandolini and Francesco Chiacchio and Cristina Trombetti}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, year={2013}, volume={145}, pages={31 - 45} }

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.

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