# Sharp Poincaré inequalities in a class of non-convex sets

### Barbara Brandolini

Università degli Studi di Napoli Federico II, Italy### Francesco Chiacchio

Università degli Studi di Napoli Federico II, Italy### Emily B. Dryden

Bucknell University, Lewisburg, USA### Jeffrey J. Langford

Bucknell University, Lewisburg, USA

## Abstract

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.

## Cite this article

Barbara Brandolini, Francesco Chiacchio, Emily B. Dryden, Jeffrey J. Langford, Sharp Poincaré inequalities in a class of non-convex sets. J. Spectr. Theory 8 (2018), no. 4, pp. 1583–1615

DOI 10.4171/JST/236