# 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}
}
• Published 7 February 2013
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics
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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