Let $\Omega$ be a uniform domain (a bounded domain satisfying the interior corkscrew and Harnack chain conditions). We study the time-independent Schrodinger operator Laplace $u + q u =0$ in $\Omega$, with $u =f$ on the boundary of Omega, where $q$ is a non-negative potential, or more generally, a locally finite Borel measure on $\Omega$, and $f$ is a non-negative function which is integrable with respect to harmonic measure on the boundary of $\Omega$. We give a sufficient condition for the existence of a non-negative solution $u$, in terms of the exponential integrability of the "Martin balayage" of $m q$, where $m$ is the minimum of 1 and the Green's function with fixed pole. The Martin balayage is defined similarly to the usual balayage (the adjoint of the Poisson integral) only with the Poisson kernel replaced with Martin's kernel. With a different constant in the exponent, this balayage condition is necessary. These results give bilateral bounds for the harmonic measure associated with the Schrodinger operator, and a criterion for the existence of the gauge function (the solution when $f =1$).
Harmonic measure for Schrodinger operators in uniform domains
Michael Frazier, University of TennesseeAuthors: Michael Frazier and Igor Verbitsky
2022 AWM Research Symposium
Analysis of Partial Differential Equations in Memory of David R. Adams