A Carleman-Type Inequality in Elliptic Periodic Homogenization

TL;DR

This paper establishes a Carleman-type inequality for solutions of elliptic equations with periodic coefficients, leading to a three-ball inequality without error terms at macroscopic scales, advancing homogenization theory.

Contribution

It introduces a new Carleman inequality in elliptic periodic homogenization and demonstrates its implications for three-ball inequalities at different scales.

Findings

Three-ball inequality without error term at macroscopic scale

Extension of inequality to any scale with doubling condition

Convergence of $H^1$-norm for solutions proven

Abstract

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients, we are interested in a Carleman-type inequality for these solutions satisfying an additional growth condition in elliptic periodic homogenization, which implies a three-ball inequality without an error term at a macroscopic scale. Moreover, if we replace the additional growth condition by the doubling condition at a macroscopic scale, then the three-ball inequality without an error term holds at any scale. The proof relies on the convergence of $H^1$-norm for the solution and the compactness argument.