Nondivergence elliptic equations with unbounded coefficients

Abstract

We study the nonvariational equation $ \sum_{i,j=1}^n a_{ij}(x)\,\frac{\partial^2 u}{\partial x_i\,\partial x_j}=f$ in domains of $r^n$. We assume that the coefficients $a_{ij}$ are in $BMO$ and the equation is elliptic, but not uniformly, and consider $f$ in $L^2(r^n)$, or even in the Zygmund class $L^2\log^\alpha L(r^n)$. We also solve Dirichlet problem.