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}(| abla u|^{p-2} abla 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.
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