TY - GEN
T1 - Quantitative propagation of smallness for solutions of elliptic equations
AU - Logunov, Alexander
AU - Laboratory, Chebyshev
AU - Malinnikova, Eugenia
N1 - 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 -