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Elliptic equations and maps of bounded length distortion. (English) Zbl 0632.35021
A discrete, open and sense-preserving map $$f: G\to R^ n$$, G a domain in $$R^ n$$, is of L-bounded length distortion (BLD) if for each path $$\alpha$$ in G, $$\ell (\alpha)/L\leq \ell (f\alpha)\leq L\ell (\alpha)$$ where $$\ell (\alpha)$$ denotes the length of $$\alpha$$. Given an elliptic, in general nonlinear, equation $$\nabla \cdot A(x,\nabla v(x))=0$$ in fG, f induces a similar equation $$\nabla \cdot f^{\#}A(x,\nabla u(x))=0$$ in G with the property that $$v\circ f$$ is a solution of $$\nabla \cdot f^{\#}A=0$$ whenever v is a solution of $$\nabla \cdot A=0$$ in fG. Thus BLD maps give new examples of generalized harmonic morphisms.
Distortion, normal families, boundary behavior and branch sets of BLD maps are studied. In particular, it is shown that $$card(f^{-1}(y))\leq L^{2n}$$ for each L-BLD map $$f: R^ n\to R^ n$$ and for each $$y\in R^ n$$. BLD maps form a subclass of quasiregular maps, see [O. Martio, S. Rickman and J. Väisälä, Ann. Acad. Sci. Fenn., Ser. A I 448, 31 p. (1971; Zbl 0223.30018)], and the invariance of solutions of special equations in the quasiregular case has been studied by Yu. G. Reshetnyak [Sib. Mat. Zh. 10, 1300-1310 (1969; Zbl 0201.098)]. In this paper different methods, based by S. Granlund, P. Lindqvist and O. Martio [Trans. Am. Math. Soc. 277, 43-73 (1983; Zbl 0518.30024)], are used.

##### MSC:
 35J60 Nonlinear elliptic equations 35J15 Second-order elliptic equations 30C62 Quasiconformal mappings in the complex plane
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##### References:
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