## Sobolev mappings with integrable dilatations.(English)Zbl 0792.30016

A continuous mapping $$f:G \to R^ n$$, $$G$$ a domain in $$R^ n$$, is said to be quasilight if for each $$y$$ the components of $$f^{-1}(y)$$ are compact. Note that every discrete mapping is quasilight. The authors show that a quasilight mapping $$f \in W^{1,n} (G)$$ satisfying $| Df(x) |^ n \leq K(x) J(x,f) \text{ a.e. for some } K \in L^ r(G),\;r>n- 1, \tag{*}$ is open and discrete. Yu. G. Reshetnyak [Sib. Math. Zh. 8, 629-658 (1967; Zbl 0162.381)] proved that for $$f \in W^{1,n} (G)$$ condition $$(*)$$ with $$K \in L^ \infty(G)$$ guarantees that $$f$$ is either constant or discrete and open. This is the fundamental result in the theory of quasiregular (space) mappings. Subsequently T. Iwaniec and V. Šverák: [Proc. Am. Math. Soc. 118, No. 1, 181-188 (1993; Zbl 0784.30015)] showed that for $$n=2$$, $$K \in L^ 1(G)$$ suffices for this conclusion. The proof of the present authors employs careful analysis of the Hausdorff dimension of $$f^{-1} (y)$$ together with some capacity estimates. The authors also show that if $$f \in W^{1,p} (G)$$, $$p \geq n+1/(n-2)$$, then the quasilight assumption is superfluous.

### MSC:

 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations

### Keywords:

integrable map; dilatation; discrete mapping

### Citations:

Zbl 0162.381; Zbl 0784.30015
Full Text:

### References:

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