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 $\left\{\begin{array}{cc}\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)\hfill & =f+\mathrm{div}\left(\right|\mathbf{F}{|}^{p-2}\mathbf{F})\text{in}\Omega ,\hfill \\ u\hfill & =g\text{on}\partial \Omega ,\hfill \end{array}\right.$ in Lorentz space, with given data $\mathbf{F}\in L^p(\Omega ;{\mathbb{R}}^n)$, $f\in L^{\frac{p}{p-1}}\left(\Omega \right)$, $g\in W^{1,p}\left(\Omega \right)$ for $1<p<\infty$ 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 (Tran, 2019; Tran and Nguyen, 2019; Tran and Nguyen, 2020; Tran and Nguyen (in press)), to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.

本文的目的是发展一类拟线性非齐次椭圆方程弱解的正则性理论，其原型为以下混合Dirichlet $p$-类型的拉普拉斯方程 $\left\{\begin{array}{cc}\mathrm{div}（|\nabla ü{|}^{p-2}\nabla ü）\hfill & =F+\mathrm{div}（|\mathbf{F}{|}^{p-2}\mathbf{F}）\text{在}\Omega ，\hfill \\ ü\hfill & =G\text{上}\partial \Omega ，\hfill \end{array}\right.$ 在洛伦兹空间中，具有给定数据 $\mathbf{F}\in {\mathrm{大号}}^p（\Omega ;{\mathbb{[R}}^{ñ}）$， $F\in {\mathrm{大号}}^{\frac{p}{p-\mathrm{1个}}}（\Omega ）$， $G\in {\mathrm{w\; ^}}^{\mathrm{1个}，p}（\Omega ）$ 对于 $\mathrm{1个}<p<\infty$ 和 $\Omega \subset {\mathbb{[R}}^{ñ}$ （$ñ\ge 2$）满足Reifenberg平坦域条件或 $p$容量均匀厚度条件，这在最近的几篇论文中都已进行了考虑。为了更好地说明我们的结果，正则性估计的证明涉及分数最大算子，并且对于带有混合数据的更一般类的拟线性非齐次椭圆方程组有效。本文不仅针对混合数据中一类更常见的问题处理了Lorentz估计，而且还改善了$\lambda$ 我们之前的工作中提出的方法（Tran，2019; Tran和Nguyen，2019; Tran和Nguyen，2020; Tran和Nguyen（印刷中）），以分数阶最大值的形式实现弱解梯度的全局Lorentz正则性估计操作员。