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

## TL;DR

This paper analyzes the behavior of free boundaries in elliptic equations with discontinuous coefficients, showing that such boundaries cannot be smooth at points of coefficient discontinuity, using integral estimates and blow-up analysis.

## Contribution

It introduces a method to classify irregular free boundary points for non-divergence elliptic equations with discontinuous coefficients using integral estimates and blow-up techniques.

## Key findings

- Free boundary points are irregular at coefficient discontinuities.
- Integral estimates help classify blow-up limits.
- Discontinuities prevent smoothness of free boundaries.

## Abstract

We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation $a_{ij}u_{ij}=u^p$, $u\ge 0$, $p\in[0, 1)$, with bounded discontinuous coefficients $a_{ij}$ having small BMO norm. We consider the simplest discontinuity of the form~$x\otimes x|x|^{-2}$ 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.