TY - GEN

T1 - Quantitative propagation of smallness for solutions of elliptic equations

AU - Logunov, Alexander

AU - Laboratory, Chebyshev

AU - Malinnikova, Eugenia

N1 - Funding Information:
A. L. was supported in part by ERC Advanced Grant 692616, ISF Grants 1380/13, 382/15 and by a Schmidt Fellowship at the Institute for Advanced Study. E. M. was supported by Project 213638 of the Research Council of Norway. MSC2010: primary 58G25; secondary 35P99.
Publisher Copyright:
© Proceedings of the International Congress of Mathematicians, ICM 2018. All rights reserved.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

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M3 - Conference contribution

AN - SCOPUS:85086310037

T3 - Proceedings of the International Congress of Mathematicians, ICM 2018

SP - 2409

EP - 2431

BT - Invited Lectures

A2 - Sirakov, Boyan

A2 - de Souza, Paulo Ney

A2 - Viana, Marcelo

PB - World Scientific Publishing Co. Pte Ltd

T2 - 2018 International Congress of Mathematicians, ICM 2018

Y2 - 1 August 2018 through 9 August 2018

ER -