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 $\mathrm{div}(|Du{|}^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally Hölder continuous. Suppose that $Du\in C_{\mathrm{loc}}^{0,\alpha }$ with $\alpha \le 1/(p-1)$, and let $\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\Phi (Du)\in C_{\mathrm{loc}}^k$ whenever $m>k/\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du{|}^{1/\alpha }$, 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.