Sharp Poincaré inequalities in a class of non-convex sets
Let $γ$ 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 $γ$, within a suitable distance $δ$ of $γ$. Denote by $μ_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $γ$ satisfies some simple geometric conditions, then $μ_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $γ$, its curvature, and $δ$. Moreover, we give explicit conditions on $δ$ that ensure $μ_1^{odd}(D)=μ_1(D)$. Finally, we can extend our bound on $μ_1^{odd}(D)$ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.