# American Institute of Mathematical Sciences

December  2018, 38(12): 6073-6090. doi: 10.3934/dcds.2018262

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

 1 School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia 2 Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom 3 Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy 4 Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133 Milan, Italy

* Corresponding author: Enrico Valdinoci

Received  July 2017 Revised  February 2018 Published  September 2018

Fund Project: Supported by INdAM Istituto Nazionale di Alta Matematica and Australian Research Council Discovery Project DP170104880 NEW Nonlocal Equations at Work

We provide an integral estimate for a non-divergence (non-varia-tional) form second order elliptic equation $a_{ij}u_{ij} = u^p$, $u≥ 0$, $p∈[0, 1)$, 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.

Citation: Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262
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##### References:
Examples of homogeneous solutions of the obstacle problem with obtuse/acute singular free boundary
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