QUANTITATIVE PROPAGATION OF SMALLNESS FOR SOLUTIONS OF ELLIPTIC EQUATIONS
Alexander Logunov, Eugenia Malinnikova · Proceedings of the International Congress of Mathematicians (ICM 2018) · 2019
Let $u$ be a solution to an elliptic equation $\text{div}(A abla u)=0$ with Lipschitz coefficients in $\mathbb{R}^n$. Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\leq 1\}$. We show that if $|u| 0, γ\in (0,1)$ 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.