| Abstract: |
| \noindent Let $\Omega$ be a convex, possibly unbounded, domain in $\mathbb{R}^2$ and denote
by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite
operator in $\Omega$. It is known that
\begin{equation}\label{1}
\mu_1(\Omega) \ge 1.
\end{equation}
The estimate is sharp since equality sign holds if $\Omega$ is any
strip.
\noindent Inequality \eqref{1} can be read as an optimal
Poincar\`e-Wirtinger inequality for functions belonging to the weighted
Sobolev space $H^1(\Omega,d\gamma_2)$, where $\gamma_2$ is the $2$%
-dimensional Gaussian measure.
\noindent We study the equality case and we show that $\mu_1(\Omega)=1$ if and only if $\Omega$ is any strip. |
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