A Local Faber-Krahn inequality and Applications to Schr\"odinger's Equation

TL;DR

This paper establishes a local Faber-Krahn inequality for solutions to Schrödinger equations, linking the potential's local integrability to the solution's maximum point and domain geometry, with applications to elliptic operators.

Contribution

It introduces a novel local inequality connecting potential bounds to solution maxima without domain shape restrictions, extending previous results in elliptic PDE analysis.

Findings

Existence of a ball with radius related to Brownian exit time where potential's norm is bounded below

Inequality holds for arbitrary domains, no geometric assumptions needed

Results extend to uniformly elliptic operators in divergence form

Abstract

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $\Delta + V$ on an arbitrary domain $\Omega$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in \Omega$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $\Omega$ with probability $\ge 1/2$. For nice (e.g., convex) domains, $T(x_0) \asymp d(x_0,\partial\Omega)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $\asymp T(x_0)^{1/2}$ such that $$ \| V \|_{L^{\frac{n}{2}, 1}(\Omega \cap B)} \ge c_n > 0, $$ provided that $n \ge 3$. In the case $n = 2$, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.