  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.

Spectral Theory;Analysis of PDEs