On the interior smoothness of solutions to second-order elliptic equations (Q2455250)

The author deals with the interior smoothness of generalized solutions to the second-order linear elliptic equation \[ \sum^n_{i,j=1} {\partial\over\partial x_i}\,\Biggl(a_{ij}(x){\partial u\over\partial x_j}\Biggr)= 0,\qquad x\in\Omega, \] where the coefficients \(a_{ij}(x)\) are measurable functions, \(\Omega\) is a given domain in \(\mathbb{R}^d\) and the symmetric matrix \(A(x)= (a_{ij}(x))\) satisfies the following inequalities \[ \gamma\leq\sum^d_{i,j=1} a_{ij}(x)\xi_i \xi_j\leq \gamma^{-1}, \] for all \(\xi= (\xi_1,\dots, \xi_d)\) with \(|\xi|= 1\) for almost all \(x\in\Omega\) and some positive \(\gamma> 0\).