We prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ of the $p$-Laplace operator ($p>1$) in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Differently from the pioneering estimate by Payne-Weinberger, our lower bound does not require any convexity assumption on $\Omega$, it involves the best isoperimetric constant relative to $\Omega$ and it is sharp, at least when $p=n=2$, as the isoperimetric constant relative to $\Omega$ goes to 0. Moreover, in a suitable class of convex planar domains, our estimate turns out to be better than the one provided by the Payne-Weinberger inequality. \noindent Furthermore, we prove that, when $p=n=2$ and $\Omega$ consists of the points on one side of a smooth curve $\gamma$, within a suitable distance $\delta$ from it, then $\mu_1(\Omega)$ can be sharply estimated from below in terms of the length of $\gamma$, the $L^\infty$ norm of its curvature and $\delta$.

Sharp bounds for Neumann eigenvalues

B. Brandolini
2018

Abstract

We prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ of the $p$-Laplace operator ($p>1$) in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Differently from the pioneering estimate by Payne-Weinberger, our lower bound does not require any convexity assumption on $\Omega$, it involves the best isoperimetric constant relative to $\Omega$ and it is sharp, at least when $p=n=2$, as the isoperimetric constant relative to $\Omega$ goes to 0. Moreover, in a suitable class of convex planar domains, our estimate turns out to be better than the one provided by the Payne-Weinberger inequality. \noindent Furthermore, we prove that, when $p=n=2$ and $\Omega$ consists of the points on one side of a smooth curve $\gamma$, within a suitable distance $\delta$ from it, then $\mu_1(\Omega)$ can be sharply estimated from below in terms of the length of $\gamma$, the $L^\infty$ norm of its curvature and $\delta$.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11588/741467
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