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

الأصل · EN

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 p-harmonic 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.