Optimal unique continuation for periodic elliptic equations on large scales

TL;DR

This paper establishes optimal large-scale doubling and three-ellipsoid inequalities for solutions of periodic elliptic equations, providing sharp estimates on the minimal scale and near-optimal constants.

Contribution

It introduces quantitatively sharp large-scale inequalities for periodic elliptic equations, improving understanding of solution behavior at large scales.

Findings

Proved optimal large-scale doubling inequality.

Established large-scale three-ellipsoid inequality.

Constants are nearly optimal, up to iterated logarithm.

Abstract

We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale on which they are valid, and are at least "almost" optimal in the prefactor constants--up to, at most, an iterated logarithm of the initial doubling ratio.