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 aijuij=up, u≥0, p∈[0,1), with bounded discontinuous coefficients aij having small BMO norm. We consider the simplest discontinuity of the form~x⊗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~aij(x).
To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.