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

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.