Sharp Poincare 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(odd)(1)(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(odd)(1)(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(odd)(1)(D) = mu(1) (D). Finally, we can extend our bound on mu(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.