Quantitative propagation of smallness for solutions of elliptic equations

TL;DR

This paper establishes quantitative estimates for how smallness in solutions of elliptic equations propagates from sets of positive measure or dimension, extending classical unique continuation results with explicit bounds.

Contribution

It provides new Remez-type inequalities and propagation of smallness estimates for solutions and their gradients of elliptic equations with Lipschitz coefficients.

Findings

Smallness propagates from sets of positive measure with explicit bounds.

Similar estimates hold for sets with Hausdorff dimension greater than n-1.

Gradient estimates extend to sets with dimension greater than n-1-c.

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 $B=\{|x|\leq 1\}$. 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.