Lorentz improving estimates for the $p$-Laplace equations with mixed data

Abstract

The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*} \begin{cases} \mathrm{div}(|\nabla u|^{p-2}\nabla u) &= f+ \ \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F}) \qquad \text{in} \ \ \Omega, \\ \hspace{1.2cm} u &=\ g \hspace{3.1cm} \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in Lorentz space, with given data $\mathbf{F} \in L^p(\Omega;\mathbb{R}^n)$, $f \in L^{\frac{p}{p-1}}(\Omega)$, $g \in W^{1,p}(\Omega)$ for $p>1$ and $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) satisfying a Reifenberg flat domain condition or a $p$-capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-$\lambda$ approach technique proposed in our preceding works~\cite{MPT2018,PNCCM,PNJDE,PNCRM}, to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.