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This paper studies a new gradient regularity in Lorentz spaces for solutionsto a class of quasilinear divergence form elliptic equations withnonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases}div(A(x,\nabla u)) &= \ div(|F|^{p-2}F) \quad \text{in} \ \ \Omega, \\\hspace{1.2cm} u &=\ \sigma \qquad \qquad \qquad \text{on} \ \ \partial \Omega.\end{cases} \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), thenonlinearity $A$ is a monotone Carath\'eodory vector valued function defined on$W^{1,p}_0(\Omega)$ for $p>1$ and the $p$-capacity uniform thickness conditionis imposed on the complement of our bounded domain $\Omega$. Moreover, forgiven data $F \in L^p(\Omega;\mathbb{R}^n)$, the problem is set up with generalDirichlet boundary data $\sigma \in W^{1-1/p,p}(\partial\Omega)$. In thispaper, the optimal good-$\lambda$ type bounds technique is applied to provesome results of fractional maximal estimates for gradient of solutions. And themain ingredients are the action of the cut-off fractional maximal functions andsome local interior and boundary comparison estimates developed in previousworks \cite{55QH4, MPT2018, MPT2019} and references therein.

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data