Sharp Poincar\'e inequalities in a class of non-convex sets

TL;DR

This paper establishes sharp lower bounds for the first nontrivial Neumann eigenvalue in certain non-convex planar and three-dimensional domains, based on geometric properties of a symmetric curve and its neighborhood.

Contribution

It provides explicit geometric conditions under which the odd Neumann eigenvalue can be sharply estimated and extends these bounds to some 3D non-convex domains.

Findings

Sharp lower bounds for ^{odd}(D) in planar domains.

Conditions ensuring ^{odd}(D) equals the standard (D).

Extension of bounds to certain three-dimensional non-convex domains.

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 $\mu_1^{odd}(D)$ 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 $\mu_1^{odd}(D)$ 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 $\mu_1^{odd}(D)=\mu_1(D)$. Finally, we can extend our bound on $\mu_1^{odd}(D)$ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.