The RadóKneserChoquet theorem for $p$harmonic mappings between Riemannian surfaces
Abstract
In the planar setting the RadóKneserChoquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of RadóKneserChoquet for $p$harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.03020
 Bibcode:
 2018arXiv180603020A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Complex Variables;
 35J47 (Primary);
 58E20;
 35J70;
 35J92 (Secondary)
 EPrint:
 38 pages, a postprint to appear in Rev. Mat. Iberoam. 36(2020), no. 6