### Abstract

Let u be a solution to an elliptic equation div(Aru) = 0 with Lipschitz coefficients in R^{n}. Assume juj is bounded by 1 in the ball B = fjxj ≥ 1g. We show that if juj < " on a set E ⊂ _{2}^{1} B with positive n-dimensional Hausdorf measure, then 1 juj ≤ C "γ on _{2} B; where C > 0; γ 2 (0; 1) do not depend on u and depend only on A and the measure of E. We specify the dependence on the measure of E in the form of the Remez type inequality. Similar estimate holds for sets E with Hausdorff dimension bigger than n = 1. For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than n = 1 = c, where c > 0 is a small numerical constant depending on the dimension only.

Original language | English (US) |
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Title of host publication | Invited Lectures |

Editors | Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana |

Publisher | World Scientific Publishing Co. Pte Ltd |

Pages | 2409-2431 |

Number of pages | 23 |

ISBN (Electronic) | 9789813272927 |

State | Published - 2018 |

Externally published | Yes |

Event | 2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil Duration: Aug 1 2018 → Aug 9 2018 |

### Publication series

Name | Proceedings of the International Congress of Mathematicians, ICM 2018 |
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Volume | 3 |

### Conference

Conference | 2018 International Congress of Mathematicians, ICM 2018 |
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Country | Brazil |

City | Rio de Janeiro |

Period | 8/1/18 → 8/9/18 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Invited Lectures*(pp. 2409-2431). (Proceedings of the International Congress of Mathematicians, ICM 2018; Vol. 3). World Scientific Publishing Co. Pte Ltd.