Nonmonotonicity of Picard principle. (English) Zbl 0598.30060

A nonnegative locally Hölder continuous function on \(0<| z| \leq 1\) is a density. The elliptic dimension of a density P(z) at \(z=0\), denoted by dim P, is the dimension of the half-module of nonnegative solutions of \(\Delta u=Pu\) on \(\Omega\) : \(0<| z| <1\) which vanish on \(| z| =1\). If dim P\(=1\), then we say that the Picard principle is valid for P. The Picard principle is valid for well-behaved densities such as \(P\in L^ 1(\Omega)\) or \(P(z)=O(| z|^{-2})\) as \(z\to 0\). In general, dim \(P\geq 1\). The authors demonstrate the rather remarkable fact that there exists densities P and Q on \(\Omega\) such that \(0\leq Q\leq P\) on \(\Omega\), and the Picard principle is valid for P but invalid for Q. This is quite contrary to one’s expectations.
Reviewer: J.L.Schiff


30F25 Ideal boundary theory for Riemann surfaces
31A10 Integral representations, integral operators, integral equations methods in two dimensions
31A35 Connections of harmonic functions with differential equations in two dimensions
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