ABSTRACT

We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators.

摘要

我们研究了在满足p容量条件的开有界非光滑域中p拉普拉斯方程的弱解梯度的加权洛伦兹空间的全局规律性。此外，梯度估计是根据与 Riesz 势和分数导数相关的分数最大算子来表示的。我们主要关注的是扩展 Tran 和 Nguyen 的结果[新梯度估计，用于解决具有一般 Dirichlet 边界数据的拟线性散度形式椭圆方程的解。J 差分方程。2020;268(4):1427–1462] 但在与 Muckenhoupt 权重相关的加权洛伦兹空间中。我们在本文中使用的关键技术是好的-λ不等式和截止分数最大算子的有趣性质。