Topological and geometrical properties of mappings with summable Jacobian in Sobolev classes. I (Q2746893)

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 \textit{Yu.~G.~Reshetnyak}, Space mappings with bounded distortion, RI: AMS, Providence (1989; Zbl 0667.30018)], in the problems of nonlinear elasticity [see, for example, \textit{T.~Iwaniec} and \textit{V.~Sverák}, Proc. Am. Math. Soc. 118, No. 1, 181-188 (1993; Zbl 0784.30015), \textit{J.~Heinonen} and \textit{P.~Koskela}, Arch. Ration. Mech. Anal. 125, No. 1, 81-97 (1993; Zbl 0792.30016), \textit{J.~Manfredi} and \textit{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:NEWLINENEWLINENEWLINE\((M_1)\) \(q\geq n-1\) for \(n=2\) and \(q>n-1\) for \(n\geq 3\);NEWLINENEWLINENEWLINE\((M_2)\) \(J(x,f)\geq 0\);NEWLINENEWLINENEWLINE\((M_3)\) \(J(x,f)\in L_{1,\text{loc}}(G)\);NEWLINENEWLINENEWLINE\((M_4)\) \(J(x,f)= 0\) if almost everywhere on a set \(A\subset G\), \(|A|>0\), then \(Df(x)=0\) almost everywhere on \(A\);NEWLINENEWLINENEWLINE\((M_5)\) \(f\:G\to\mathbb R^n\) is continuous;NEWLINENEWLINENEWLINE\((M_6)\) \(f\:G\to\mathbb R^n\) possesses at least one of the following properties:NEWLINENEWLINENEWLINE\((a)\) the mapping is almost absolutely continuous;NEWLINENEWLINENEWLINE\((b)\) the adjugate matrix \(\text{adj} Df(x)\) (i.e., \(Df(x) \text{adj} Df(x)=J(x,f) \text{Id}\)) belongs to \(L_{q,\text{loc}}\), \(q=n/(n-1)\).NEWLINENEWLINENEWLINEThe main result is as follows:NEWLINENEWLINENEWLINETheorem 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 \(f\)NEWLINENEWLINENEWLINE\((1)\) belongs to \(W^1_{q,\text{loc}}\) with \(q=np/(p+1)\);NEWLINENEWLINENEWLINE\((2)\) is open and discrete;NEWLINENEWLINENEWLINE\((3)\) is differentiable almost everywhere on \(G\) in the classical sense.NEWLINENEWLINENEWLINEThe 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 [\textit{S.~K.~Vodop'yanov} and \textit{A.~D.~Ukhlov}, Sib. Math. J. 37, No. 1, 62-78 (1996; Zbl 0870.43005) and \textit{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.