Regularity for $H$-systems of bounded distortion

Abstract

We consider weak solutions $u\in W^{1,n}(U,{\mathbf{R}}^{n+1})$ of the $H$-system $-\mathrm{div}(|\mathrm{D}u{|}^{n-2}\mathrm{D}u)=(H\circ u)\mathbf{J}u$, 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 $|\mathrm{D}u{|}^{n-2}\mathrm{D}u\circ \mathrm{D}{u}^{\ast }$ by $u$ to a linear functional on continuous compactly supported maps of ${\mathbf{R}}^{n+1}$ into $\mathrm{Hom}({\mathbf{R}}^{n+1},{\mathbf{R}}^{n+1})$, 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 $|\mathrm{D}u{|}^n{\mathcal{L}}^n$ by $u$, and these imply boundedness.