A Faber-Krahn inequality with drift

Abstract

Let $Ω$ be a bounded $C^{2,α}$ domain in $\R^n$ ($n\geq 1$, $0<α<1$), $Ω^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $Ω$, $τ\geq 0$ and $v\in L^{\infty}(Ω,\R^n)$ with $\left\Vert v\right\Vert\_{\infty}\leq τ$. If $λ\_{1}(Ω,τ)$ denotes the principal eigenvalue of the operator $-Δ+v\cdot\nabla$ in $Ω$ with Dirichlet boundary condition, we establish that $λ\_{1}(Ω,v)\geq λ\_{1}(Ω^{\ast},τe\_{r})$ where $e\_{r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $Ω=Ω^{\ast}$ and $v=τe\_{r}$. 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.