Quantitative propagation of smallness for solutions of elliptic equations

Abstract

Let $u$ be a solution to an elliptic equation $\text{div}(A\nabla u)=0$ with Lipschitz coefficients in ${\mathbb{R}}^n$. Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\le 1\}$. We show that if $|u|<\epsilon$ on a set $E\subset \frac{1}{2}B$ with positive $n$-dimensional Hausdorf measure, then $$|u|\le C{\epsilon }^{\gamma }\text{ on }\frac{1}{2}B,$$ where $C>0,\gamma \in (0,1)$ 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.