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Distortion of quasiconformal mappings with identity boundary values. (English) Zbl 1307.30056

For a proper subdomain \(D\) of \(\mathbb{R}^n\), \(n \geq 2\), let \(\mathrm{Id}_K(\partial D)\) be the class of all \(K\)-quasiconformal mappings \(f : \mathbb{R}^n \rightarrow \mathbb{R}^n\) such that \(f(x) = x\) for every \(x \in \mathbb{R}^n \setminus D\). Several authors, among other O. Teichmüller [Deutsche Math. 7, 336–343 (1944; Zbl 0060.23401)] and J. Krzyz [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 16, 99–101 (1968; Zbl 0184.31001)], have studied the deviation of \(f \in\mathrm{Id}_K(\partial D)\) from the identity map for various domains \(D\) in the plane and in the space. Krzyź’s result for the class \(\mathrm{Id}_K(\partial B^2)\) is extended to convex domains \(D\) in \(\mathbb{R}^n\) and the result is \(j_D(x,f(x)) \leq \log(1+(c_{K,n} -1)^{1/2})\), where \(c_{K,n}\) has an explicit dependence on \(K\) and \(n\) in terms the capacity of the Teichmuller ring domain and \(j_D(x,y)\) stands for the distance-ratio metric \(\log(1+|x-y|/\min(d(x, \partial D),d(y, \partial D)))\). The constant \(c_{K,n}\) satisfies \(c_{K,n} \rightarrow 1\) as \(K \rightarrow 1\). The proof uses sharp modulus estimates. The method leads to several explicit distortion estimates in terms of the quasihyperbolic metric \(k_D\), for example \(k_{B^n}(x,f(x)) \leq 4(1+\log 6)^{1/2 }(K-1)^{1/2 }\) for \(f \in Id_K(\partial B^n)\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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