论文标题
Lorentz改进了$ p $ laplace方程的估计，并带有混合数据
Lorentz improving estimates for the $p$-Laplace equations with mixed data
论文作者
论文摘要
本文的目的是为一类准线性非均匀椭圆方程开发规律性理论，其原型是以下混合的dirichlet $ p $ -p $ -laplace type \ begin的{align*}} \ begin {align*} \ begin {case}}}} {case}} \ mathrm {div}（div}（div）（ \ Mathrm {div}（| \ Mathbf {f} |^{p-2} \ Mathbf {f}）\ qquad \ text {in} \ \ \ \ \ \ \ \ \ \ \ \ hspace {1.2cm {1.2cm} \ end {cases} \ end {align*}在lorentz空间中，并带有给定数据$ \ mathbf {f} \ in l^p（ω; \ mathbb {r}^n）$，$ f \ in l^{\ frac {p} {p} {p-1}} {p-1}} {p-frac} { $ p> 1 $和$ω\ subset \ mathbb {r}^n $（$ n \ ge 2 $）满足Reifenberg平面域状态或$ p $ - 容量 - 容量统一厚度条件，这在最近的几篇论文中都考虑到。为了更好地指定我们的结果，规律性估计的证明涉及分数最大运算符，并且有效，对具有混合数据的更通用的准线性非均匀椭圆方程式有效。 This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-$λ$ 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.
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} \ \ Ω, \\ \hspace{1.2cm} u &=\ g \hspace{3.1cm} \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz space, with given data $\mathbf{F} \in L^p(Ω;\mathbb{R}^n)$, $f \in L^{\frac{p}{p-1}}(Ω)$, $g \in W^{1,p}(Ω)$ for $p>1$ and $Ω\subset \mathbb{R}^n$ ($n \ge 2$) 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-$λ$ 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.
