Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations

Abstract

In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally Hölder continuous. Suppose that $Du\in C^{0,α}_{\rm loc}$ with $α\le 1/(p-1)$, and let $Φ$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $Φ(Du)\in C^k_{\rm loc}$ whenever $m>k/α$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/α}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate Hölder assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.