JournalsjstVol. 8, No. 4pp. 1583–1615

Sharp Poincaré inequalities in a class of non-convex sets

  • Barbara Brandolini

    Università degli Studi di Napoli Federico II, Italy
  • Francesco Chiacchio

    Università degli Studi di Napoli Federico II, Italy
  • Emily B. Dryden

    Bucknell University, Lewisburg, USA
  • Jeffrey J. Langford

    Bucknell University, Lewisburg, USA
Sharp Poincaré inequalities in a class of non-convex sets cover
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Abstract

Let γ\gamma be a smooth, non-closed, simple curve whose image is symmetric with respect to the yy-axis, and let DD be a planar domain consisting of the points on one side of γ\gamma, within a suitable distance δ\delta of γ\gamma. Denote by μ1odd(D)\mu_1^{\textup{odd}}(D) the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the yy-axis. If γ\gamma satisfies some simple geometric conditions, then μ1odd(D)\mu_1^{\mathrm{odd}}(D) can be sharply estimated from below in terms of the length of γ\gamma, its curvature, and δ\delta. Moreover, we give explicit conditions on δ\delta that ensure μ1odd(D)=μ1(D)\mu_1^{\mathrm{odd}}(D)=\mu_1(D). Finally, we can extend our bound on μ1odd(D)\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.

Cite this article

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. 1583–1615

DOI 10.4171/JST/236