### Abstract

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 ∈ [0, 1), with bounded discontinuous coefficients a_{ij} 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 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.

Original language | English |
---|---|

Title of host publication | Discrete and Continuous Dynamical Systems - Series A |

Pages | 6073-6090 |

Number of pages | 18 |

Volume | 38 |

Edition | 12 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

### Publication series

Name | Discrete and Continuous Dynamical Systems- Series A |
---|---|

Publisher | Southwest Missouri State University |

ISSN (Print) | 1078-0947 |

### Fingerprint

### Cite this

*Discrete and Continuous Dynamical Systems - Series A*(12 ed., Vol. 38, pp. 6073-6090). (Discrete and Continuous Dynamical Systems- Series A). https://doi.org/10.3934/dcds.2018262

}

*Discrete and Continuous Dynamical Systems - Series A.*12 edn, vol. 38, Discrete and Continuous Dynamical Systems- Series A, pp. 6073-6090. https://doi.org/10.3934/dcds.2018262

**Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients.** / Dipierro, Serena; Karakhanyan, Aram; Valdinoci, Enrico.

Research output: Chapter in Book/Conference paper › Conference paper

TY - GEN

T1 - Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

AU - Dipierro, Serena

AU - Karakhanyan, Aram

AU - Valdinoci, Enrico

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Blow-up sequences

KW - Free boundary

KW - Monotonicity formulae

KW - Non-divergence operators

UR - http://www.scopus.com/inward/record.url?scp=85054321656&partnerID=8YFLogxK

U2 - 10.3934/dcds.2018262

DO - 10.3934/dcds.2018262

M3 - Conference paper

VL - 38

T3 - Discrete and Continuous Dynamical Systems- Series A

SP - 6073

EP - 6090

BT - Discrete and Continuous Dynamical Systems - Series A

ER -