Quantitative unique continuation for elliptic equations with H\"older continuous potentials

Abstract

We study quantitative unique continuation for second order elliptic equations with lower-order terms of H\"older regularity via a weighted frequency function method. We establish quantitative three-ball inequalities and corresponding vanishing-order bounds for Schr\"odinger equations with H\"older potentials and H\"older gradient terms, and corresponding results for elliptic equations with variable leading coefficients. Our results are quantitative with explicit dependence of H\"older norms in the three-ball inequalities. These fill in the gap for quantitative unique continuation between bounded potentials and $C^1$ potentials.