Regularity for $H$-systems of bounded distortion

We consider weak solutions $u\in W^{1,n}(U,{\mathbf{R}}^{n+1})$ of the $H$-system $-\mathrm{div}(\mathrm{\mid }\mathrm{D}u{\mathrm{\mid }}^{n-2}\mathrm{D}u)=(H\circ u)\mathbf{J}u$, where $n\ge 2$, $U$ is the open unit ball in ${\mathbf{R}}^n$, and $H$ is bounded. We show that $u$ is locally bounded if its distortion $\mathrm{\mid }\mathrm{D}u{\mathrm{\mid }}^n\mathrm{/}\mathrm{\mid }\mathbf{J}u\mathrm{\mid }$ is bounded on the set where $\mathrm{\mid }u\mathrm{\mid }$ is large. If moreover $H$ is Hölder continuous, then $u$ is locally Hölder continuous. The proof pushes forward $\mathrm{\mid }\mathrm{D}u{\mathrm{\mid }}^{n-2}\mathrm{D}u\circ \mathrm{D}{u}^{\ast }$ by $u$ to a linear functional on continuous maps of ${\mathbf{R}}^{n+1}$ into $\mathrm{Hom}({\mathbf{R}}^{n+1},{\mathbf{R}}^{n+1})$ with compact support, whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael–Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati, and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of $\mathrm{\mid }\mathrm{D}u{\mathrm{\mid }}^n{\mathcal{L}}^n$ by $u$, and these imply boundedness.