## 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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### References:

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