Rendiconti Lincei - Matematica e Applicazioni


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Volume 27, Issue 4, 2016, pp. 443–464
DOI: 10.4171/RLM/743

Published online: 2016-10-04

The equality case in a Poincaré–Wirtinger type inequality

Barbara Brandolini[1], Francesco Chiacchio[2], David Krejčiřík[3] and Cristina Trombetti[4]

(1) Università degli Studi di Napoli Federico II, Italy
(2) Università degli Studi di Napoli Federico II, Italy
(3) Czech Technical University in Prague, Czech Republic
(4) Università degli Studi di Napoli Federico II, Italy

It is known that, for any convex planar set $\Omega$, the first non-trivial Neumann eigenvalue $\mu_1 (\Omega)$ of the Hermite operator is greater than or equal to 1. Under the additional assumption that $\Omega$ is contained in a strip, we show that $\mu_1 (\Omega) = 1$ if and only if $\Omega$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.

Keywords: Hermite operator, Neumann eigenvalues, thin strips

Brandolini Barbara, Chiacchio Francesco, Krejčiřík David, Trombetti Cristina: The equality case in a Poincaré–Wirtinger type inequality. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 27 (2016), 443-464. doi: 10.4171/RLM/743