Second order elliptic equations whose coefficients have their first derivatives weakly-\(L^ n\)
Let \(a_{ij}(x)\) \(W^{1,n}(G)\), G open bounded subset of \({\mathbb{R}}^ n\), be such that \(| \xi |^ 2\leq \sum_{i,j}a_{ij}(x)\xi_ i\xi_ j\leq m| \xi |^ 2\), suppose moreover that \[ \text{[spherical rearrangement of }\sum_{i}(\sum_{j}\frac{\partial a_{ij}(x)}{\partial x_ j})^ 2]<\frac{B^ 2}{| x|^ 2},\quad B<\frac{n-2}{2}, \] \[ \text{[spherical rearrangement of }\sum_{irs}(\sum_{k} \frac{\partial}{\partial x_ k}(a_{ir}a_{ks}- a_{ik}a_{rs}))^ 2]\leq \] \[ \leq D^ 2/| x|^ 2,\quad D<\frac{n-2}{2}2^{-1/n}; \] thenthe problem \[ \sum_{i,j}a_{ij}(x)u_{x_ ix_ j}=f\quad in\quad G,\quad u|_{\partial G}=0 \] has a unique solution in \(W^{2,2}(G)\cap W_ 0^{1,2}(G)\) \(\forall f\in L^ 2(G)\).
scientific article; zbMATH DE number 3927463