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Extremal mappings of finite distortion. (English) Zbl 1089.30013

In this paper the authors initiate the study of extremal problems for mappings of finite distortion. A mapping \(f \in W_{\text{loc}}^{1,1}(\Omega, \Omega')\) between subdomains \(\Omega\) and \(\Omega'\) of \(\mathbb R^n\) is of finite distortion if there is a measurable function \(1 \leq K(x) < \infty\) such that \[ | Df(x)| ^n \leq K(x) J(x, f) \] where \(D\) is the linear differential map and \(J(x, f)\) is the Jacobian of \(f\) at \(x\). The smallest such function is called the outer distortion, and is defined by \[ K(x, f) = \frac{| Df(x)| ^n}{J(x, f)}. \] A typical extremal problem is the following. Let \(\mathcal F\) be some family of mappings of finite distortion on a domain \(\Omega\). Given some function \(f_o \in \mathcal F\), find a mapping \(f \in \mathcal F\) such that \(f = f_o\) on \(\partial \Omega\) and such that the integral \[ \int_\Omega K(x, f) \] is minimized. The problem is studied both for various families \(\mathcal F\) and for different distortion functions \(K\). The families \(\mathcal F\) are usually families of homeomorphisms in \(W_{\text{loc}}^{1,1}(\Omega, \Omega')\) for particular domains \(\Omega\) and \(\Omega'\), while examples of \(K\) include, in addition to outer distortion, inner distortion and generalized outer distortion where the ordinary outer distortion function is composed with a strictly increasing convex function.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35J15 Second-order elliptic equations
35J70 Degenerate elliptic equations
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