Quantitative propagation of smallness for solutions of elliptic equations

Alexander Logunov, Chebyshev Laboratory, Eugenia Malinnikova

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationInvited Lectures
EditorsBoyan Sirakov, Paulo Ney de Souza, Marcelo Viana
PublisherWorld Scientific Publishing Co. Pte Ltd
Pages2409-2431
Number of pages23
ISBN (Electronic)9789813272927
StatePublished - 2018
Externally publishedYes
Event2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil
Duration: Aug 1 2018Aug 9 2018

Publication series

NameProceedings of the International Congress of Mathematicians, ICM 2018
Volume3

Conference

Conference2018 International Congress of Mathematicians, ICM 2018
CountryBrazil
CityRio de Janeiro
Period8/1/188/9/18

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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  • Cite this

    Logunov, A., Laboratory, C., & Malinnikova, E. (2018). Quantitative propagation of smallness for solutions of elliptic equations. In B. Sirakov, P. N. de Souza, & M. Viana (Eds.), Invited Lectures (pp. 2409-2431). (Proceedings of the International Congress of Mathematicians, ICM 2018; Vol. 3). World Scientific Publishing Co. Pte Ltd.