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

TL;DR

This paper develops quantitative unique continuation results for elliptic equations with H"older continuous potentials, providing explicit bounds and inequalities that bridge the gap between bounded and smoother potentials.

Contribution

It introduces a weighted frequency function approach to establish three-ball inequalities and vanishing-order bounds for elliptic equations with H"older regularity, extending prior results.

Findings

Established quantitative three-ball inequalities for H"older potentials

Derived explicit vanishing-order bounds with H"older norm dependence

Extended unique continuation results to less regular 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.