Some aspects of the theory of stability of conformal mappings (Q1363996)

Let \(n\) be an natural number, \(n\geq 2\), \(H\) be an Euclidean space \(\mathbb{R}^m \), \(n<m <+ \infty\). Let \(G^1= \bigcup_{\Delta \subset \mathbb{R}^n} \{g: \Delta\to H\}\) be a union of all conformal mappings \(g:\Delta \to H\), where \(\Delta\) is a domain in \(\mathbb{R}^n\) and let \[ W^1 = W^{1,n,m} =\bigcup_{\Delta \subset \mathbb{R}^n} W^1(\Delta) =\bigcup_{\Delta \subset \mathbb{R}^n} \bigcup_{p>n} W^1_{p,loc} (\Delta, \mathbb{R}^m), \] where \(W^1_{p,loc} (\Delta, \mathbb{R}^m)\) is a Sobolev space of functions \(f:\Delta \to\mathbb{R}^m\) that have generalized first order derivatives that are locally summable. In this paper, among others, the following result is proved. Theorem. There exists a nonnegative real function \(\beta: [0,1/2] \times(0,1) \to[0,1]\) such that a) if \(\rho \in (0,1)\), then \(\beta ({\mathcal E}, \rho)\to \beta (0,\rho) =0\) when \({\mathcal E} \to 0\); and b) if \(f: \Delta \to \mathbb{R}^m\), \(\Delta \subset \mathbb{R}^n\), \(m>n\geq 2\) is a mapping from a class \(W^1 (\Delta)\), and \(\Xi (f,G^1) \leq {\mathcal E} <1/2\), then for each number \(\rho \in(0,1)\) and each bounded domain \(\Delta'\) which is contained in the domain \(\Delta\) together with its \(\tau\)-neighbourhood \(U_\tau (\Delta') =\{x\in \mathbb{R}^n: \inf_{t\in \Delta'} |x- t|< \tau\}\), where \(\tau = {1-\rho \over 2 \rho}\) diam \(\Delta'\), there exists a mapping \(g: \Delta' \to\mathbb{R}^m\) from a class \(G'\) such that \[ \bigl|f(x)- g(x)\bigr|\leq\beta ({\mathcal E}, \rho) \text{ diam }f \bigl( U_\tau (\Delta') \bigr),\;x\in \Delta', \] where \(\Xi(f,G^1)\) is a generalized solution to the following differential inequalities: \[ \bigl|df(x) \bigr|^n \leq(1+ {\mathcal E}) \bigl|J(x,f) \bigr|. \]