Boundary quantitative unique continuation for solutions of elliptic equations

TL;DR

This paper establishes boundary quantitative unique continuation estimates for elliptic equations with Neumann boundary conditions on compact manifolds, using Carleman estimates to derive boundary doubling inequalities.

Contribution

It introduces a novel approach combining local and global Carleman estimates to handle regularity issues in boundary unique continuation problems.

Findings

Derived boundary doubling inequalities for elliptic equations

Addressed regularity challenges in boundary value problems

Extended unique continuation results to boundary settings

Abstract

We study the quantitative unique continuation on the boundary for solutions of elliptic equations with Neumann boundary conditions for bounded potentials and boundary potentials on compact manifolds with boundary. The boundary doubling inequality is derived from the combination of local Carleman estimates and global Carleman estimates. Some special attentions are paid to overcome the regularity issues arising from this boundary value problem.