In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form −div(𝒜(x,∇u)) = μinΩ,u = 0on∂Ω, where μ is a finite signed Radon measure in Ω, Ω ⊂ ℝn is a bounded domain such that its complement ℝn∖Ω is uniformly p-thick and 𝒜 is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case 1 < p ≤ 3n−2 2n−1. Our result extends the earlier results [19, 22] to the strongly singular case 1 < p ≤ 3n−2 2n−1 and a recent result [18] by considering rough conditions on the domain Ω and the nonlinearity 𝒜.

中文翻译：

---

具有强奇异情况的测量数据的拟线性方程的全局洛伦兹梯度估计：1 < p ≤ 3n−2 2n−1

在本文中，我们研究了洛伦兹空间中拟线性椭圆方程解梯度的全局正则性估计，测量数据形式为 -div(𝒜(X,∇你)) = μ在Ω,你 = 0在∂Ω, 在哪里μ是有限符号氡测量Ω,Ω ⊂ ℝn是一个有界域，使得它的补码ℝn∖Ω是一致的p-厚而厚𝒜是满足强奇异情况的增长和单调性条件的 Carathéodory 向量值函数1 < p ≤ 3n-2 2n-1. 我们的结果将早期的结果 [19, 22] 扩展到强奇异情况1 < p ≤ 3n-2 2n-1和最近的结果 [18] 通过考虑域上的粗略条件Ω和非线性𝒜.