Sharp Poincaré inequalities in a class of non-convex sets

Abstract

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^{\textup{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^{\mathrm{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^{\mathrm{odd}}(D)=\mu_1(D) . Finally, we can extend our bound on \mu_1^{\mathrm{odd}}(D) to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.