The geometry of planar p-harmonic mappings: convexity, level curves and the isoperimetric inequality

We discuss various representations of planar p-harmonic systems of equations and their solutions. For coordinate functions of p-harmonic maps we analyze signs of their Hessians, the Gauss curvature of p-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar pharmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.