In the planar setting the Radó-Kneser-Choquet 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 Radó-Kneser-Choquet 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.
- Pub Date:
- June 2018
- Mathematics - Analysis of PDEs;
- Mathematics - Complex Variables;
- 35J47 (Primary);
- 35J92 (Secondary)
- 38 pages, a postprint to appear in Rev. Mat. Iberoam. 36(2020), no. 6