Quantitative uniqueness for elliptic equations at the boundary of $C^{1, Dini}$ domains

TL;DR

This paper establishes an optimal boundary vanishing order bound for solutions to variable coefficient Schrödinger equations in $C^{1,Dini}$ domains, providing a quantitative form of strong unique continuation at the boundary.

Contribution

It introduces a boundary analogue of interior unique continuation results using a frequency function approach for variable coefficient equations.

Findings

Optimal upper bound on vanishing order at boundary

Quantitative strong unique continuation at boundary

Extension of interior results to boundary setting

Abstract

Based on a variant of the frequency function approach of Almgren, we establish an optimal upper bound on the vanishing order of solutions to variable coefficient Schr\"odinger equations at a portion of the boundary of a $C^{1,Dini}$ domain. Such bound provides a quantitative form of strong unique continuation at the boundary. It can be thought of as a boundary analogue of an interior result recently obtained by Bakri and Zhu for the standard Laplacian.