Regularity for quasilinear elliptic equations in metric measure spaces
Simon Schulz, Ivan Yuri Violo
TL;DR
This work develops a robust regularity theory for quasilinear elliptic equations $\mathrm{div}(\Psi(|\nabla u|)\nabla u)=f$ on metric measure spaces with synthetic lower Ricci curvature bounds, proving second-order and Lipschitz regularity via a Galerkin approach using Laplacian eigenfunctions. The authors establish strong a priori estimates, including a local $L^2$-Laplacian bound, weighted second-order (Caccioppoli) estimates, and $L^{\infty}$ gradient bounds, accommodating both nondegenerate and $p$-growth regimes. They introduce a truncation/regularization strategy and a Galerkin scheme to transfer Euclidean second-order regularity to the non-smooth setting, and they prove convergence results to pass to the original operator $\mathrm{div}(\Psi(|\nabla u|)\nabla u)=f$. A Cheng–Yau type gradient bound for $p$-harmonic functions is derived, and corollaries cover minimal surfaces and eigenfunctions within the $RCD(K,N)$ framework. The results substantially extend Hölder theory by providing quantitative second-order and Lipschitz regularity in a broad, non-smooth geometric context.
Abstract
In the present article we prove second-order and Lipschitz regularity for quasilinear elliptic equations in metric spaces endowed with a lower bound on the Ricci curvature. The estimates we obtain are quantitative and cover a large class of elliptic equations with polynomial growth. As a particular case we settle the Lipschitz regularity of $p$-harmonic functions for all values of $p\in(1,\infty)$, proving also a Cheng-Yau type inequality. These results are the first in this setting that simultaneously address a wide family of elliptic operators and extend beyond the classical Hölder regularity theory. Our strategy rests on the use of Galerkin's method, which we employ as an alternative to the traditional difference quotients technique.