Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

created by difazio on 05 Oct 2021

[BibTeX]

preprint

Inserted: 5 oct 2021

Year: 2018

ArXiv: 1810.12496 PDF

Abstract:

We study regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F$ in bounded domains in $\R^n$. The vector field $\A$ 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 $\A$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\calM^{q,\lambda}$ or weighted space $L^q_w$ whenever $\F ^{\frac{1}{p-1}}$ is respectively in $\calM^{q,\lambda}$ or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{\frac{q}{p}}$. 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 $\F ^{\frac{1}{p-1}}$ belongs to.

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