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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