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 odd 1 (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 odd 1 (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 odd 1 (D) = µ1(D). Finally, we can extend our bound on µ 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.
Journal of Spectral Theory
Dryden, Emily; Langford, Jeffrey; Brandolini, Barbara; and Chiacchio, Francesco. "Sharp Poincaré inequalities in a class of non-convex sets." Journal of Spectral Theory (2018) : 1583-1615.