Second order elliptic equations whose coefficients have their first derivatives in Lorentz spaces (Q1071943)

The author studies the following Dirichlet problem: \[ Lu=\sum a_{ij} u_{x_ i} u_{x_ j}=f(x)\quad in\quad G,\quad u=0\quad on\quad \partial G. \] G is an open bounded subset of \({\mathbb{R}}^ n\) with \(\partial G\) sufficiently smooth, L is uniformly elliptic, \(a_{ij}(x)\) are bounded measurable functions and \(f\in L^ 2(G)\). In addition all the derivatives \(D_{x_ h}a_{ij}(x)\) are assumed to be in L(n,p), the (n,p) Lorentz space. An existence and uniqueness theorem in \(W^{2,2}(G)\cap W_ 0^{1,2}(G)\) is proved.