Revista Matemática Iberoamericana

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Published online first: 2020-02-12
DOI: 10.4171/rmi/1178

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

Giuseppe Di Fazio[1] and Truyen Nguyen[2]

(1) Università degli Studi di Catania, Italy
(2) University of Akron, USA

We study regularity for solutions of quasilinear elliptic equations of the form $\mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F}$ in bounded domains in $\mathbb{R}^n$. The vector field $\mathbf{A}$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\mathbf{A}$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\mathcal{M}^{q,\lambda}$ or weighted space $L^q_w$ whenever $|\mathbf{F}|^{1/(p-1)}$ is respectively in $\mathcal{M}^{q,\lambda}$ or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{q/p}$. In addition, our two-weight estimate allows the possibility to acquire the regularity for $\nabla u$ in a weighted Morrey space that is different from the functional space that the data $|\mathbf{F}|^{1/(p-1)}$ belongs to.

Keywords: Elliptic equations, $p$-Laplacian, gradient estimates, Calderón–Zygmund estimates, weighted Morrey spaces, regularity, two-weight inequality, Muckenhoupt class

Di Fazio Giuseppe, Nguyen Truyen: Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations. Rev. Mat. Iberoam. Electronically published on February 12, 2020. doi: 10.4171/rmi/1178 (to appear in print)