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Lewy’s theorem fails in higher dimensions. (English) Zbl 0711.31003
In Bull. Am. Math. Soc. 42, 689-692 (1936; Zbl 0015.15903), H. Lewy showed that, if $$n=2$$, a one-to-one harmonic map of $${\mathbb{R}}^ n$$ to itself has non-vanishing Jacobian. The author gives a simple example to show that Lewy’s theorem fails for $$n\geq 3$$.
Reviewer: J.C.Wood

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 35J45 Systems of elliptic equations, general (MSC2000) 58E20 Harmonic maps, etc.
##### Keywords:
harmonic map; non-vanishing Jacobian; Lewy’s theorem
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