To the problem of determining a mapping from its normalized Jacobi matrix
Let \(D\) be a domain of \(\mathbb{R}^n\) \((n \geq 2)\) and \(f : D \to\mathbb{R}^n\) a mapping, whose coordinate functions \(f^i\) \((i = 1, \dots, n)\) belong to the Sobolev space \(W^1_{n, \text{loc}} (D)\), its Jacobi matrix \(f'\) is defined a.e. in \(D\) and \(J(f,x) = \text{det} f' (x)\). \(f\) is said to be a mapping with bounded distortion if it is continuous and satisfies the system of differential equations \[ f'(x) = K(x) |J(x,f) |^{1/n}, \tag{1} \] a.e in \(D\), where \(\text{det} K(x)\) does not change its sign in \(D\), \(|\text{det} K(x) |= 1\) a.e. in \(D\) and there exists a constant \(C \geq 1\) such that \(|K |_{\infty, D} \leq C\). The author, in a previous paper [Sib. Mat. Zh. 33, 53-61 (1992; Zbl 0806.26008)], established some necessary conditions on the normalized Jacobi matrix \(K(x) = f'(x) \{|J(x,f) |^{1/n}\}\) (considered as a quasiconformal characteristic) if \(f \in C^3\) with \(J(x,f) \neq 0\). In the paper under review, the author proves that these conditions still hold for the more general case of mappings with bounded distortion. He obtains also an existence theorem for the solutions of system (1), allowing to obtain the mapping \(f\) from \(K\). Finally, he shows that the existence of a constant \(K_0\) such that \(|K(x) |\leq K_0\) a.e. in \(D\) implies that \(f\) is locally homeomorphic, hence, on account of a result of \textit{V. A. Zorich} [Mat. Sb., N. Ser. 74(116), 417-433 (1967; Zbl 0181.08701)], if \(D=\mathbb{R}^n,f\) is quasiconformal in \(\mathbb{R}^n\).
scientific article; zbMATH DE number 825557
scientific article