The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of Spectral Theory


Full-Text PDF (806 KB) | Metadata | Table of Contents | JST summary
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