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

Abstract

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.

to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XIV, Issue 2, June 2015

Cited by in corpus (2)

• On asymptotically sharp bi-Lipschitz inequalities of quasiconformal mappings satisfying inhomogeneous polyharmonic equations
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