\(n\)-harmonic morphisms in space are Möbius transformations
A mapping \(f\) of an open subset of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\) is called a \(p\)-harmonic morphism if \(u \circ f\) is \(p\)-harmonic in the preimage \(f^{-1}(d)\) whenever \(u\) is \(p\)-harmonic in \(D\). Recall that \(u\) is \(p\)-harmonic in an open set \(\Omega\) if \(u \in C(\Omega) \cap W^{1,p}(\Omega)\) satisfies the \(p\)-Laplace equation \[ \text{div}(| \nabla u|^{p - 2} \nabla u)=0 \] in \(\Omega\); here \(1 < p < \infty\). The case \(p = 2\) is the classical Laplacian; then harmonic morphisms are known to be analytic or antianalytic mappings in the planar case, and similarity transformations (i.e. compositions of rotations and translations) in higher dimensions, and vice versa. For a general \(p\), it is easily seen that the \(p\)-Laplacian is invariant under similarity transformations and, moreover, the \(n\)-Laplacian is invariant under all 1-quasiregular mappings (i.e. Möbius maps if \(n \geq 3\)). Therefore these mappings are \(p\)-harmonic morphisms. In the present paper the authors show that a discrete \(p\)-harmonic morphism necessarily is 1-quasiregular if \(p = n\) and a similarity if \(p \neq n\). The paper complements the paper [\textit{J. Heinonen, T. Kilpelaeinen} and \textit{O. Martio}, Nagoya Math. J. 125, 115-140 (1992; Zbl 0776.31007)], where harmonic morphisms between nonlinear harmonic spaces are characterized.
scientific article; zbMATH DE number 575626