A Faber-Krahn inequality with drift

Francois Hamel (1) , Nikolai Nadirashvili (1) , Emmanuel Russ (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\R^n$ ($n\geq 1$, $0<\alpha<1$), $\Omega^{\ast}$ be the open Euclidean ball centered at $0$ having the same Lebesgue measure as $\Omega$, $\tau\geq 0$ and $v\in L^{\infty}(\Omega,\R^n)$ with $\left\Vert v\right\Vert_{\infty}\leq \tau$. If $\lambda_{1}(\Omega,\tau)$ denotes the principal eigenvalue of the operator $-\Delta+v\cdot\nabla$ in $\Omega$ with Dirichlet boundary condition, we establish that $\lambda_{1}(\Omega,v)\geq \lambda_{1}(\Omega^{\ast},\tau e_{r})$ where $e_{r}(x)=x/\left\vert x\right\vert$. Moreover, equality holds only when, up to translation, $\Omega=\Omega^{\ast}$ and $v=\tau e_{r}$. This result can be viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian.

Mots clés: Faber-Krahn; isoperimetric inequality; eigenvalue; drift

Domaines: Equations aux dérivées partielles [math.AP]

Francois Hamel, Nikolai Nadirashvili, Emmanuel Russ. A Faber-Krahn inequality with drift. 2006. ⟨hal-00087322⟩