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

Abstract

We provide an integral estimate for a non-divergence (non-varia-tional) form second order elliptic equation \begin{document}$a_{ij}u_{ij} = u^p$\end{document} , \begin{document}$u≥ 0$, $p∈[0, 1)$\end{document} , with bounded discontinuous coefficients \begin{document}$a_{ij}$\end{document} having small BMO norm. We consider the simplest discontinuity of the form \begin{document}$x\otimes x|x|^{-2}$\end{document} at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when \begin{document}$p = 0$\end{document} ) cannot be smooth at the points of discontinuity of \begin{document}$a_{ij}(x)$\end{document} . 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.