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Journal of Spectral Theory

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Volume 8, Issue 4, 2018, pp. 1583–1615
DOI: 10.4171/JST/236

Published online: 2018-10-22

Sharp Poincaré inequalities in a class of non-convex sets

Barbara Brandolini[1], Francesco Chiacchio[2], Emily B. Dryden[3] and Jeffrey J. Langford[4]

(1) Università degli Studi di Napoli Federico II, Italy
(2) Università degli Studi di Napoli Federico II, Italy
(3) Bucknell University, Lewisburg, USA
(4) Bucknell University, Lewisburg, USA

Let $\gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $\gamma$, within a suitable distance $\delta$ of $\gamma$. Denote by $\mu_1^{\textup{odd}}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $\gamma$ satisfies some simple geometric conditions, then $\mu_1^{\mathrm{odd}}(D)$ can be sharply estimated from below in terms of the length of $\gamma$, its curvature, and $\delta$. Moreover, we give explicit conditions on $\delta$ that ensure $\mu_1^{\mathrm{odd}}(D)=\mu_1(D)$. Finally, we can extend our bound on $\mu_1^{\mathrm{odd}}(D)$ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.

Keywords: Neumann eigenvalues, lower bounds, non-convex domains

Brandolini Barbara, Chiacchio Francesco, Dryden Emily, Langford Jeffrey: Sharp Poincaré inequalities in a class of non-convex sets. J. Spectr. Theory 8 (2018), 1583-1615. doi: 10.4171/JST/236