Local bounds for nonlinear higher-order vector fields for the p-Laplace equation

Abstract

We study higher regularity for weak solutions of the $p$-Laplace equation $-\hat{I}{”}_{p}u=f$ in a domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ for $p$ sufficiently close to 2. For $m\ge 3$, assuming that $f$ satisfies suitable Sobolev and Hölder regularity conditions, we prove that the nonlinear quantity $|\nabla u{|}^{m-2}\nabla u$ belongs to $W_{loc}^{m-1,q}(\hat{I}\text{c◯})$, and that $|\nabla u{|}^{m-2}{D}^{2}u$ belongs to $W_{loc}^{m-2,q}(\hat{I}\text{c◯})$, for any $q\ge 2$. Furthermore, we obtain uniform $L^{\infty }$ bounds for the weighted $(m-1)$-th derivatives of $|\nabla u{|}^{m-2}\nabla u$ and the weighted $(m-2)$-th derivatives of $|\nabla u{|}^{m-2}{D}^{2}u$, providing quantitative control even near critical points of $\nabla u$.