Positive solutions of Schrödinger equations and fine regularity of boundary points

Abstract

Given a Lipschitz domain $\Omega$ in ${\mathbb{R}}^N$ and a nonnegative potential $V$ in $\Omega$ such that $V(x)d(x,\partial \Omega {)}^2$ is bounded in $\Omega$ we study the fine regularity of boundary points with respect to the Schrödinger operator $L_V:=\Delta -V$ in $\Omega$. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z\in \partial \Omega$ are established. The main result is a simple (explicit if $\Omega$ is smooth) necessary and sufficient condition involving the size of $V$ for $z$ to be finely regular. An essential intermediate result consists in a majorization of ${\int }_A|\frac{u}{d(.,\partial \Omega )}{|}^{2}dx$ for $u$ positive harmonic in $\Omega$ and $A\subset \Omega$. Conditions for almost everywhere regularity in a subset $A$ of $\partial \Omega$ are also given as well as an extension of the main results to a notion of fine ${\mathcal{L}}_1|{\mathcal{L}}_0$-regularity, if ${\mathcal{L}}_j=\mathcal{L}-V_j$, $V_0,V_1$ being two potentials, with $V_0\le V_1$ and $\mathcal{L}$ a second order elliptic operator.

version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1 is corrected

References in corpus (1)

• Inverse Problem for a Curved Quantum Guide

Cited by in corpus (2)

• Hopf potentials for the Schrödinger operator
• The Hopf lemma for the Schrödinger operator