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Harmonic morphisms in nonlinear potential theory. (English) Zbl 0776.31007

For fixed \(1<p< \infty\), let \(A_ p\) denote the family of all mappings \(a:\mathbb{R}^ n \times \mathbb{R}^ n \to \mathbb{R}^ n\), \((x,h) \to a(x,h)\), satisfying appropriate conditions (such as being comparable to \(| h |^{p-1})\). A function \(u\) defined in an open set \(\Omega \subset \mathbb{R}^ n\) is said to be \(a\)-harmonic in \(\Omega\) if it is a continuous (generalized) solution in \(\Omega\) of the quasilinear elliptic equation \(-\text{div} a(x,\nabla u)=0\).
Definition: Let \(a^*\) and \(a\) belong to \(A_ p\). A continuous mapping \(f:\Omega \to \mathbb{R}^ n\) is an \((a^*,a)\)-harmonic morphism if \(u \circ f\) is \(a^*\)-harmonic in \(f^{-1}(\Omega)\) whenever \(u\) is \(a\)-harmonic in \(\Omega\). \(f\) is said to be an \(A_ p\)-harmonic morphism if \(f\) is an \((a^*,a)\)-harmonic morphism for some \(a^*\) and \(a\) in \(A_ p\).
Definition: A continuous mapping \(f:\Omega \to \mathbb{R}^ n\) is \(K\)-quasi- regular if it belongs to the Sobolev space \(W^{1,n}_{\text{loc}}(\Omega)\) and satisfies the inequality \(| f'(x)|^ n\leftrightharpoons KJ_ f(x)\) a.e. in \(\Omega\), where \(f'(x)\) is the formal derivative matrix and \(J_ f(x)\) is the Jacobian determinant.
This paper is an extensive study of \(A_ p\)-harmonic morphisms especially in the cae \(p=n\) where the authors prove that every sense- preserving \(A_ n\)-harmonic morphism is quasi-regular. Also, if \(1<p<n\) and \(f:\Omega \to \mathbb{R}^ n\) is a (suitably restricted) \(A_ p\)- harmonic morphism, then \(f\) is of bounded length distortion in every compact set of \(\Omega\). There is a section concerning asymptotic estimates of \(a\)-harmonic functions near an essential isolated singularity. There are quite a few references in the text to closely related work upon which the present proofs depend.

MSC:

31C45 Other generalizations (nonlinear potential theory, etc.)
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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