Regularity of weak solutions of $n$-dimensional H-systems

Abstract

We use the method of Strzelecki [Calc. Var. 1 (2003)] to generalize the Bethuel theorem [C. R. Acad. Sci. Paris 314 (1992)] to $n$-dimensional H-systems. We prove that if $u$ is a parameterization of an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, weakly satisfies the system $\Delta_n u = H(u) u_{x_1} \times \cdots \times u_{x_n}$ and additionally has $n-1$ weak derivatives in $L^{n/(n-1)}$, then $u$ is Hölder continuous. Furthermore it is continuous up to the boundary, whenever it has continuous trace. We also give an example showing that the structure of the H-system is relevant and that the assumption that $u$ has $n-1$ weak derivatives does not trivialize the problem.