Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain Ω in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}^N}$$\end{document} and a nonnegative potential V in Ω such that V(x) d(x, ∂Ω)2 is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator LV := Δ − V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${z \in \partial \Omega}$$\end{document} . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${z \in \partial \Omega}$$\end{document} to be finely regular. An intermediate result consists in a majorization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$$\end{document} for u positive harmonic in Ω and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \subset \Omega}$$\end{document}. Conditions for almost everywhere regularity in a subset A of ∂Ω are also given as well as an extension of the main results to a notion of fine \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ L}_1 \vert \mathcal{L}_0}$$\end{document}-regularity, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$$\end{document} being two potentials, with V0 ≤ V1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}$$\end{document} a second order elliptic operator.