Sharp Poincaré inequalities in a class of non-convex sets

Let 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 within a suitable distance ı of . Denote by odd 1 .D/ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the y-axis. If satisfies some simple geometric conditions, then odd 1 .D/ can be sharply estimated from below in terms of the length of , its curvature, and ı. Moreover, we give explicit conditions on ı that ensure odd 1 .D/ D 1 .D/. Finally, we can extend our bound on odd 1 .D/ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex