Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation a(ij)u(ij) = u(p), u >= 0, p is an element of [0, 1), with bounded discontinuous coefficients a(ij) having small BMO norm. We consider the simplest discontinuity of the form x circle times x|x|(-2) at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when p = 0) cannot be smooth at the points of discontinuity of a(ij) (x).