Let be a smooth, non-closed, simple curve whose image is symmetric with respect to the -axis, and let be a planar domain consisting of the points on one side of , within a suitable distance of . Denote by the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the -axis. If satisfies some simple geometric conditions, then can be sharply estimated from below in terms of the length of , its curvature, and . Moreover, we give explicit conditions on that ensure . Finally, we can extend our bound on to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.
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Barbara Brandolini, Francesco Chiacchio, Emily B. Dryden, Jeffrey J. Langford, Sharp Poincaré inequalities in a class of non-convex sets. J. Spectr. Theory 8 (2018), no. 4, pp. 1583–1615DOI 10.4171/JST/236