A Faber-Krahn inequality with drift

Abstract

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 $e\_{r}(x)=x/| x|$. 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.