The p-harmonic approximation and the regularity of p-harmonic maps

We extend to the degenerate case p = 2, Simon’s approach to the classical regularity theory of harmonic maps of Schoen & Uhlenbeck, by proving a “p-Harmonic Approximation Lemma". This allows to approximate functions with p-harmonic functions in the same way as the classical harmonic approximation lemma (going back to De Giorgi) does via harmonic functions. Finally, we show how to combine this tool with suitable regularity estimates for solutions to degenerate elliptic systems with a critical growth right hand side, in order to obtain partial C^{1,\alpha}-regularity of p-harmonic maps.