Positive solutions of Schrödinger equations and fine regularity of boundary points

Abstract

Given a Lipschitz domain $Ω$ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $Ω$ such that $V(x)\, d(x,\partial Ω)^2$ is bounded in $Ω$ we study the fine regularity of boundary points with respect to the Schrödinger operator $L_V:= Δ-V$ in $Ω$. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z \in \partial Ω$ are established. The main result is a simple (explicit if $Ω$ 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 $\int_A | {\frac {u} {d(.,\partial Ω)}} | ^2\, dx$ for $u$ positive harmonic in $Ω$ and $A \subset Ω$. Conditions for almost everywhere regularity in a subset $A $ of $ \partial Ω$ 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 $V_0 \leq V_1$ and ${\mathcal L}$ a second order elliptic operator.