Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

Abstract

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ Ω, \mbox{ and } \ u = g \ \text{on} \ \partial Ω, \end{align*} that related to Riesz potentials. Here, in our setting, $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity $A$. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

30 pages