Regularity for $H$-systems of bounded distortion

Abstract: We consider weak solutions $u \in W<sup>{1,n}(U,\mathbf{R}<sup>{n+1})$ of the $H$-system $$- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}<sup>n$, and $H$ is bounded. We show that $u$ is locally bounded if its distortion $|\mathrm{D} u|ⁿ / |\mathbf{J} u|$ is bounded on the set where $|u|$ is large. If moreover $H$ is Hölder continuous, then $u$ is locally Hölder continuous. The proof pushes forward $$|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u<sup>*$$ by $u$ to a linear functional on continuous maps of $\mathbf{R}<sup>{n+1}$ into $$\operatorname{Hom}(\mathbf{R}<sup>{n+1},\mathbf{R}<sup>{n+1})$$ 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 $|\mathrm{D} u|ⁿ \mathscr{L}<sup>n$ by $u$, and these imply boundedness.