Schrodinger equations with very singular potentials in Lipschitz domains

TL;DR

This paper investigates Schrödinger operators with highly singular potentials in Lipschitz domains, establishing estimates for harmonic functions, Green potentials, and solutions, extending previous results from smooth to Lipschitz domains.

Contribution

It provides new estimates for solutions and potentials of Schrödinger operators with singular potentials in Lipschitz domains, generalizing prior smooth domain results.

Findings

Derived estimates for positive harmonic functions and Green potentials.

Extended results to Lipschitz domains from smooth domain cases.

Provided conditions for existence of ground states and Hardy constants.

Abstract

Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second condition that guarantees the existence of a ground state $\Phi_V$. If, for example, $V>0$ this condition reads $1<c_H(V)$ (= the Hardy constant relative to $V$). We derive estimates of positive $L_V$ harmonic functions and of positive Green potentials of measures $\tau\in {\mathfrak M}_+(\Omega;\Phi_V)$. These imply estimates of positive $L_V$ supersolutions and of $L_V$ subsolutions. Similar results have been obtained in [7] in the case of smooth domains.