A Faber-Krahn inequality with drift

Abstract

Let $\Omega$ be a bounded $C^{2,\alpha }$ domain in ${\mathbb{R}}^n$ ($n\ge 1$, $0<\alpha <1$), ${\Omega }^{\ast }$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $\Omega$, $\tau \ge 0$ and $v\in L^{\infty }(\Omega ,{\mathbb{R}}^n)$ with $\Vert v\Vert \_\infty \le \tau$. 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 $\lambda \_1(\Omega ,v)\ge \lambda \_1({\Omega }^{\ast },\tau e\_r)$ 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.

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