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Uniform limits of certain \(A\)-harmonic functions with applications to quasiregular mappings. (English) Zbl 0727.35022

Let \(u_ 1,u_ 2,...,u_ m\) be nonconstant uniform limits (on compact subsets) of A-harmonic functions in \(\{\) x: \(| x| <R\}\subset {\mathbb{R}}^ n\) where A satisfies certain elliptic structure conditions. The authors show that if there exists \(\lambda\geq 0\) such that (i) \(\{\) x: \(u_ i(x)<-\lambda \}\cap \{x:\) \(u_ j(x)<-\lambda \}=\emptyset\), (ii) \(| u^+_ j-u^+_ i| \leq \lambda\), and (iii) \(| u_ j(0)| \leq \lambda\), for \(1\leq i,j\leq m\), then \(m\leq c\) where c depends only on the structure conditions and n. As an application they show that their theorem provides a completely P.D.E. proof of Rickman’s generalization of Picard’s theorem to quasiregular mappings.

MSC:

35B99 Qualitative properties of solutions to partial differential equations
30C62 Quasiconformal mappings in the complex plane
31C45 Other generalizations (nonlinear potential theory, etc.)
35J60 Nonlinear elliptic equations
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