Positive solutions and harmonic measure for Schr\"{o}dinger operators in uniform domains

TL;DR

This paper establishes bilateral estimates for positive solutions of Schrödinger equations in uniform domains, providing conditions for existence and bounds of solutions related to harmonic measure and Green's functions.

Contribution

It introduces new bilateral bounds and necessary conditions for solutions of Schrödinger operators in uniform domains, linking harmonic measure, Green's functions, and exponential integrability.

Findings

Bilateral pointwise estimates for positive solutions

Necessary and sufficient conditions for solution existence

Criteria for the gauge function in Schrödinger equations

Abstract

We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = \omega u \, \,& & \mbox{in} \, \, \Omega, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial \Omega , \end{aligned} \right. \end{equation*} in a bounded uniform domain $\Omega\subset {\bf R}^n$, where $\omega$ is a locally finite Borel measure in $\Omega$, and $f\ge 0$ is integrable with respect to harmonic measure $d H^{x}$ on $\partial\Omega$. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of $M^{*} (m \omega)(z)=\int_\Omega M(x, z) m(x)\, d \omega (x)$ on $\partial\Omega$ with respect to $f \, d H^{x_0}$, where $M(x, \cdot)$ is Martin's function with pole at $x_0\in \Omega, m(x)=\min (1, G(x, x_0))$, and $G$ is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schr\"{o}dinger operator $-\triangle - \omega $ on $\Omega$, and in the case $f=1$, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.