In this paper we consider a linear elliptic equation in divergence form

$$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$(0.1)

Assuming the coefficients \(a_{ij}\) in \(W^{1,n}(\Omega )\) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution \(u\in L^{n'}_\mathrm{loc}(\Omega )\) of (0.1) is actually a weak solution in \(W^{1,2}_\mathrm{loc}(\Omega )\).

中文翻译：

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关于散度形式的线性椭圆型方程的极弱解的正则性

在本文中，我们考虑发散形式的线性椭圆方程

$$ \ begin {aligned} \ sum _ {i，j} D_j（a_ {ij}（x）D_i u）= 0 \ quad \ hbox {in} \ Omega。\ end {aligned} $$（0.1）

假设系数\（A_ {IJ} \）在\（W ^ {1，N}（\欧米茄）\）与满足一定迪尼型连续性状态连续性的模量，我们证明了任何非常弱溶液\（ u \ in（0.1）的L ^ {n'} _ \ mathrm {loc}（\ Omega）\）实际上是\（W ^ {1,2} _ \ mathrm {loc}（\ Omega）的弱解\）。