Positive solutions and harmonic measure for Schrödinger operators in uniform domains

Michael W. Frazier, Igor E. Verbitsky

Published 2020-11-08Version 1

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.