6533b835fe1ef96bd129ec18
RESEARCH PRODUCT
An optimal Poincaré-Wirtinger inequality in Gauss space
Cristina TrombettiFrancesco ChiacchioAntoine HenrotBarbara Brandolinisubject
Hermite operatorHermite polynomialsGeneral Mathematics010102 general mathematicsGaussMathematics::Spectral TheorySpace (mathematics)Gaussian measure01 natural sciencesOmega35B45; 35P15; 35J70CombinatoricsSobolev spaceSettore MAT/05 - Analisi Matematica0103 physical sciencesDomain (ring theory)[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Neumann eigenvaluesharp bounds010307 mathematical physics0101 mathematicsSign (mathematics)Mathematicsdescription
International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-01-01 | Mathematical Research Letters |