CLASSIFICATION OF IRREGULAR FREE BOUNDARY POINTS FOR NON-DIVERGENCE TYPE EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

摘要

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).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.