This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A₂ class of Muckenhoupt weights. Global and interior weighted W^1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W^1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Hölder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

中文翻译：

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拟线性退化椭圆型方程弱解的加权Calderón-Zygmund估计

本文研究规律性估计在Sobolev空间一类的形式的div [A（退化奇异准线性椭圆问题的弱解X，ü，∇ Ù）] = DIV [ ˚F ]与非均匀狄利克雷边界条件有界的非光滑域。系数A可以是x的奇异值，也可以是简并的，或者在x上都可以表现为类似于权重函数μ，这在Muckenhoupt权重的A ₂类中。整体和内部加权W ^1，p（Ω，ω对带有其他权重函数ω的这些方程式的弱解建立了正则性估计。对于μ = 1而言，由于依赖于A的解u而获得的结果甚至是新的。在线性方程式的情况下，我们的W ^1，p正则性估计值可以视为Sobolev对应于Hölder正则性估计值的建立由Fabes，CE Kenig和RP Serapioni撰写。