Boundary behavior of positive solutions of \(\Delta u=Pu\) on a Riemann surface. (English) Zbl 0940.31003

Satō, Takeyoshi

J. Math. Soc. Japan 51, No. 1, 167-179 (1999).

On a hyperbolic Riemann surface \(R\), a \(P\)-solution is a \(C^2\)-function satisfying the Schrödinger equation \(\Delta u=Pu\) where \(P dx dy\) is a nonnegative Hölder continuous 2-form on \(R\). Let \(\Delta_1\) be the minimal Martin boundary and for \(b\in \Delta_1\) let \({\mathcal G}(b)\) be the associated filter; let \(\chi\) be the canonical measure of the constant function 1.
The author proves a Fatou-type limit theorem for positive \(P\)-solutions: There is a set of positive \(\chi\)-measure \(\Delta_{HP}^0 \subset \Delta_1\) such that a positive continuous \(P\)-supersolution on \(R\) has a limit following the filter \({\mathcal G}(b)\) on \(\Delta_{HP}^0\), \(\chi\) a.e.; under a restriction on \(P\), it is shown that \(\Delta_{HP}^0\) is almost as large as \(\Delta_1\).

Reviewer: Victor Anandam (Riyadh)

MSC:

31A35	Connections of harmonic functions with differential equations in two dimensions
31C35	Martin boundary theory

Keywords:

Schrödinger equation on a hyperbolic Riemann surface; fine limit; Martin boundary