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 this, one also shows that

$$\sup_{\rho < 1/2, p\in B_{1/2}(0)} \rho^{-\alpha} \int_{B_\rho(p)} |\Delta u|$$

is bounded. This implies that $f\in\mathcal{H}^1$. I just include this extra detail in case this question fits into a general framework of optimal interior regularity for Poisson's equation on a disk when the right hand side is Hardy. (This is the reason for the earlier form of my question.)

Can anybody help?