Take the 2-minute tour ×  Context: I am currently reading
through the freely available lecture notes from Tristan Riviere
(here) on the applicability of integration by compensation in
the analysis of various geometrically motivated PDEs.  I have
attempted to find something in the vast literature to the
following effect: suppose $u:D^2\rightarrow\mathbb{R}^n$, $u\in
W^{1,2}$, $\Delta u = f(u,\nabla u) =
2H\partial_xu\times\partial_yu$. Then  $$u\in C^{0,\alpha}(D')
\rightarrow u\in C^\infty(D''),$$  where $D'' \subset\subset D'
\subset\subset D$.  In other words, an interior estimate. With
the little regularity on hand, it appears to be very difficult.
I find this surprising because most of the time, proving the
Hölder regularity of the solution is the 'most difficult' part.
I have the feeling that I am missing an obvious reference or a
well-known folklore argument.  Showing the Hölder continuity of
the solution relies on deriving a Morrey-type estimate with the
help of the Wente lemma. In the process of doing