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Topological and geometrical properties of mappings with summable Jacobian in Sobolev classes. I. (Russian, English) Zbl 0983.30009

Sib. Mat. Zh. 41, No. 1, 23-48 (2000); translation in Sib. Math. J. 41, No. 1, 19-39 (2000).
The goal of the paper is to indicate analytic conditions on the mappings \(f\:G\to\mathbb R^n\), \(G\subset\mathbb R^n\), \(n\geq 2\), of the Sobolev classes which guarantee certain topological properties (openness, discreteness, differentiability) for \(f\). The mapping \(f\) is open if the image of an open set is open; and \(f\) is discrete if the inverse image \(f^{-1}(y)\) of every point \(y\in\mathbb R^n\) consists of isolated points. Analytic constraints on \(f\) are convenient to be reading as the requirement of finiteness of some norms of local distortion \(K(x)=|Df(x)|^n/J(x,f)<\infty\), where \(J(x,f)\) is the Jacobian and \(|Df(x)|\) is the norm of the gradient \(Df(x)\). The necessity of studying the topological properties of the mappings arises in the theory of mappings with bounded distortion [see Yu. G. Reshetnyak, Space mappings with bounded distortion, RI: AMS, Providence (1989; Zbl 0667.30018)], in the problems of nonlinear elasticity [see, for example, T. Iwaniec and V. Sverák, Proc. Am. Math. Soc. 118, No. 1, 181-188 (1993; Zbl 0784.30015), J. Heinonen and P. Koskela, Arch. Ration. Mech. Anal. 125, No. 1, 81-97 (1993; Zbl 0792.30016), J. Manfredi and E. Villamor, Bull. Am. Math. Soc., New Ser. 32, No. 2, 235-240 (1995; (1998; Zbl 0857.30020)]. In the article under review, the author obtains some topological results for mappings \(f\in W^1_{q,\text{loc}}(G)\) under the following constraints:
\((M_1)\) \(q\geq n-1\) for \(n=2\) and \(q>n-1\) for \(n\geq 3\);
\((M_2)\) \(J(x,f)\geq 0\);
\((M_3)\) \(J(x,f)\in L_{1,\text{loc}}(G)\);
\((M_4)\) \(J(x,f)= 0\) if almost everywhere on a set \(A\subset G\), \(|A|>0\), then \(Df(x)=0\) almost everywhere on \(A\);
\((M_5)\) \(f\:G\to\mathbb R^n\) is continuous;
\((M_6)\) \(f\:G\to\mathbb R^n\) possesses at least one of the following properties:
\((a)\) the mapping is almost absolutely continuous;
\((b)\) the adjugate matrix \(\text{adj} Df(x)\) (i.e., \(Df(x) \text{adj} Df(x)=J(x,f) \text{Id}\)) belongs to \(L_{q,\text{loc}}\), \(q=n/(n-1)\).
The main result is as follows:
Theorem 1. Suppose that \(f\:G\to\mathbb R^n\), \(G\subset\mathbb R^n\), \(n\geq 2\), is a nonconstant mapping of the class \(W^1_{1,\text{loc}}\) which satisfies \((M_2)\)–\((M_6)\) and that \(K(x)\in L_{p,\text{loc}}\) for some \(n-1\leq p\leq \infty\) if \(n=2\) and \(n-1<p\leq \infty\) if \(n\geq 3\). Then \(f\)
\((1)\) belongs to \(W^1_{q,\text{loc}}\) with \(q=np/(p+1)\);
\((2)\) is open and discrete;
\((3)\) is differentiable almost everywhere on \(G\) in the classical sense.
The methods of the article are based on the change-of-variable formula with multiplicity function and degree of mapping and develop the methods of the papers [S. K. Vodop’yanov and A. D. Ukhlov, Sib. Math. J. 37, No. 1, 62-78 (1996; Zbl 0870.43005) and S. K. Vodop’yanov, Sib. Math. J. 37, No. 6, 1113-1136 (1996; Zbl 0876.30020)]. In the case of \(p=n\), the author obtains a new proof of the openness and discreteness for mappings with bounded distortion which does not use approximation of a mapping by smooth mappings. Theorem 1 also covers Reshetnyak’s theorem and the results of the above-cited papers by T. Iwaniec and V. Sverák, J. Heinonen and P. Koskela, and J. Manfredi and E. Villamor.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
43A80 Analysis on other specific Lie groups
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