%0 Journal Article
%T The geometry of planar $p$-harmonic mappings: convexity, level curves and the isoperimetric inequality
%A Tomasz Adamowicz
%J Mathematics
%D 2013
%I arXiv
%R 10.2422/2036-2145.201201_010
%X 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.
%U http://arxiv.org/abs/1309.6113v1