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 $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}^n$, 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. By a result of Fusco, Kristensen, Leone, and Verde, it follows that $u$ is locally Hölder continuous if $H$ is continuous, and continuously differentiable with locally Hölder continuous derivative if $H$ is locally Lipschitz. The proof pushes forward $|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u^*$ by $u$ to a linear functional on continuous compactly supported maps of $\mathbf{R}^{n+1}$ into $\operatorname{Hom}(\mathbf{R}^{n+1},\mathbf{R}^{n+1})$, 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 $|\mathrm{D} u|^n \mathscr{L}^n$ by $u$, and these imply boundedness.