  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.

Analysis of PDEs