@inproceedings{0a8ae06be37c4b6eb162a6210ed79895,

title = "Quantitative propagation of smallness for solutions of elliptic equations",

abstract = "Let u be a solution to an elliptic equation div(Aru) = 0 with Lipschitz coefficients in Rn. Assume juj is bounded by 1 in the ball B = fjxj ≥ 1g. We show that if juj < {"} on a set E ⊂ 21 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.",

author = "Alexander Logunov and Chebyshev Laboratory and Eugenia Malinnikova",

year = "2018",

language = "English (US)",

series = "Proceedings of the International Congress of Mathematicians, ICM 2018",

publisher = "World Scientific Publishing Co. Pte Ltd",

pages = "2409--2431",

editor = "Boyan Sirakov and {de Souza}, {Paulo Ney} and Marcelo Viana",

booktitle = "Invited Lectures",

address = "Singapore",

note = "2018 International Congress of Mathematicians, ICM 2018 ; Conference date: 01-08-2018 Through 09-08-2018",

}