Quantitative propagation of smallness for solutions of elliptic equations

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 $B=\{|x|\leq 1\}$. We show that if $|u| < \varepsilon$ on a set $E \subset \frac{1}{2} B$ with positive $n$-dimensional Hausdorf measure, then $$|u|\leq C\varepsilon^\gamma \text{ on } \frac{1}{2}B,$$ 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.