Harmonic mappings on Grushin planes, part II

Abstract: We study harmonic mappings between the source domain in the $α$-Grushin plane and the target domain in the $β$-Grushin plane for $α$ not necessarily equal to $β$ and $α\in [0,1)$. Our harmonic mappings arise as a Euler--Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.