## 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

### MSC:

 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

### Keywords:

elliptic dimension; Picard principle
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### References:

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