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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.

35J60 Nonlinear elliptic equations
35J15 Second-order elliptic equations
30C62 Quasiconformal mappings in the complex plane
Full Text: DOI EuDML
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