Positive solutions and harmonic measure for Schrödinger operators in uniform domains

Abstract

We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = ωu \, \,& & \mbox{in} \, \, Ω, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial Ω, \end{aligned} \right. \end{equation*} in a bounded uniform domain $\Omega \subset {\mathbf{R}}^n$, 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 $M^{\ast }(m\omega )(z)={\int }_{\Omega }M(x,z)m(x)d\omega (x)$ on $\partial \Omega$ with respect to $fd{H}^{x_0}$, 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 Schrödinger operator $-△-\omega$ 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.

38 pages

References in corpus (1)

• Global estimates for kernels of Neumann series and Green's functions