Liouville type theorems for mappings with bounded (co)-distortion
Let \(M\) and \(N\) be oriented \(n\)-dimensional Riemannian manifolds. A mapping \(f:M\to N\) has bounded \(s\)-distortion \((1\leq s<\infty)\) if \(f\in W^{1,1}_{\text{loc}} (M,N)\) and \(|df_x|^s \leq KJ_f(x)\) a.e. \(x \in M\). This paper deals with a number of questions for such kinds of mappings. The main result is the following Liouville type theorem: Theorem: Let \(M\) and \(N\) be oriented \(n\)-dimensional Riemannian manifolds and let \(f\in W^{1, 1}_{\text{loc}}(M,N)\) be a mapping with bounded \(s\)-distortion with \(s>(n-1)\). Assume that \(M\) is \(p\)-parabolic, where \(1/p+(n-1)/s=1\). Then either \(f\) is constant a.e. or \(N\) is also \(p\)-parabolic.
scientific article; zbMATH DE number 1883509
scientific article