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Stability of classes of quasiregular mappings in several spatial variables. (English. Russian original) Zbl 0856.30019

Sib. Math. J. 36, No. 1, 43-54 (1995); translation from Sib. Mat. Zh. 36, No. 1, 47-59 (1995).
Let \((\mathbb{R}^n)^k = \mathbb{R}^n \times \mathbb{R}^n \times \cdots \times \mathbb{R}^n\), \((k\)-times), \(x \in (\mathbb{R}^n)^k\) may be written as \(x = (x_1, x_2, \dots, x_k)\) where \(x_j \in \mathbb{R}^n \), \(j = 1,2, \dots, k\). For a given domain \(U \subset (\mathbb{R}^n)^k\) we introduce a function \(f = (f_1, f_2, \dots, f_m) : U \to (\mathbb{R}^n)^m\) of the class \(W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\) with Jacobi matrix \[ f'(x) = \bigl[ f_{ij}'(x) \bigr]_{i = 1, 2, \dots, m; j = 1, 2, \dots, k} \quad \text{where} \quad \bigl[ f_{ij}' \bigr] = \left[ {\partial f^i \over \partial x_j} \right] \in \mathbb{R}^{n \times n} \] are the matrices of a dimension \(n \times n\). Definition A mapping \(f : U \to (\mathbb{R}^n)^m\) is called the quasiregular mapping of several spatial variables if \(1^\circ\) \(f \in W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\); \(2^\circ\) there exists a constant \(K < \infty\) such that \[ \sum_{i,j} |f_{ij}' |^n \leq K n^{n/2} \cdot \sum_{i,j} \text{det} f_{ij}', \quad \text{where} \quad |A |= \bigl |[a_{rs}] \bigr |= \Bigl( \sum_{r,s} a^2_{rs} \Bigr)^{1/2} \] and let \(\text{det} A\) denote the determinant of \(A\). The constant \(K = K(f)\) is called a coefficient of quasiregularity of a mapping \(f\) and the set of all quasiregular mappings such that \(K(f) \leq K\) is denoted by \(G(K) = G^m_{n,k} (K)\). In this paper some properties of such classes of quasiregular mappings of several spatial variables are investigated. This results in the special cases coincide with the earliest results of the author and the results of Kopylov and Reshetnyak.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26E10 \(C^\infty\)-functions, quasi-analytic functions
26B10 Implicit function theorems, Jacobians, transformations with several variables
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References:

[1] N. S. Dairbekov, ”Quasiregular mappings of severaln-dimensional variables,” Sibirsk. Mat. Zh.,34, No. 4, 87–102 (1993). · Zbl 0809.30014
[2] N. S. Dairbekov, ”The concept of a quasiregular mapping of severaln-dimensional variables,” Dokl. RAN,324, No. 3, 511–514 (1992).
[3] A. P. Kopylov, Stability in theC-Norm of Classes of Mappings [in Russian], Nauka, Novosibirsk (1990). · Zbl 0772.30023
[4] A. P. Kopylov, ”On stability of classes of holomorphic maps in several variables. I. The concept of stability. Liouville’s theorem,” Sibirsk. Mat. Zh.,23, No. 2, 83–111 (1982). · Zbl 0509.32012
[5] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982). · Zbl 0487.30011
[6] Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis [in Russian], Nauka, Novosibirsk (1982). · Zbl 0523.53025
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