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.


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