Quasiconformal maps and Poincaré domains
Let \(D\) be a domain in \(\mathbb{R}^m\) and \(1\leq p<\infty\). If for some constant \(C <\infty\) \(\inf_{a\in\mathbb{R}} |u- a|_{L^p(D)} \leq C|\nabla u|_{L^p(D)}\) for all \(u \in C^1(D)\), then \(D\) is called a \(p\)-Poincaré domain, Poincaré domains are invariant under locally bi-Lipschitz maps [\textit{R. Hurri}: Ann. Acad. Sci. Fenn., Ser. A I Math. Dissertationes 71, 1-42 (1988; Zbl 0667.46023)] but not necessarily under quasiconformal maps. Some extra conditions are needed: Let \(D\) be a John domain in \(\mathbb{R}^n\) [\textit{O. Martio}, \textit{J. Sarvas}: Ann. Acad. Sci. Fenn., Ser. AI 4, 383-401 (1979; Zbl 0427.30021)] and \(f\) a quasiconformal map of \(D\) onto \(D^*\) with (*) \(|f'|_{L^{n+ \varepsilon} (D)}< \infty\). Then \(D^*\) is \(p\)-Poincaré for \(p\geq n\) and for \(p_0<p<n\) where \(p_0\) essentially depends on the higher integrability condition (*). The method of the proof gives explicit bounds for the Poincaré constant \(C\) for \(D^*\).
scientific article; zbMATH DE number 1036051
scientific article