Gradient estimates for singular \(p\)-Laplace type equations with measure data (Q6582319)

The authors investigate the regularity of weak solutions of the quasilinear elliptic equation \(-\mathrm{div}\, A(x,\nabla u) = \mu\), where \(\mu\) is a locally finite signed Radon measure in a domain \(\Omega \subset \mathbb{R}^n\), and the vector field \(A\) satisfies a set of assumptions with \(|\nabla u|^{p-2} \nabla u\) as a prototype. In the case \(p \in (1,2)\), the authors obtain interior and global pointwise and Lipschitz gradient estimates, as well as the continuity of the gradient. In particular, the results extend those obtained in [\textit{F. Duzaar} and \textit{G. Mingione}, J. Funct. Anal. 259, No. 11, 2961--2998 (2010; Zbl 1200.35313); \textit{Q.-H. Nguyen} and \textit{N. C. Phuc}, J. Funct. Anal. 278, No. 5, Article ID 108391, 35 p. (2020; Zbl 1437.35123)].