Gradient estimates of general nonlinear singular elliptic equations with measure data
Gradient estimates of general nonlinear singular elliptic equations with measure data
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DOI：
10.1016/j.jde.2023.07.003
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发表时间：
2023
影响因子：
2.4
通讯作者：
Junjie Zhang;Shenzhou Zheng;Zhaosheng Feng
中科院分区：
文献类型：
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作者：
Junjie Zhang;Shenzhou Zheng;Zhaosheng Feng
在齐次Dirichlet边界条件下，给出了一般非线性奇异椭圆型方程−div A(x，u，Du)=μ重整化解的梯度的整体Calderón-Zygmund估计，其中μ是有限符号Radon测度.对于奇异情况p∈(1，2−1/n]，关联的非线性表现为关于Du的椭圆p-拉普拉斯，其x变量中的不连续性是用小BMO来度量的，并且要求关于u变量的Lipschitz连续性。首先利用扰动技巧和加权Vitaly型覆盖方法建立了加权Good-λ型不等式，然后在加权Lorentz空间和Lorentz-Morrey空间中证明了期望的全局梯度估计。作为直接推论，我们最终得到了加权Orlicz空间中的全局梯度正则性。
We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations− div A (x, u, D u)= μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1, 2− 1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.