Quantitative unique continuation for parabolic equations with Neumann boundary conditions

TL;DR

This paper develops a quantitative unique continuation estimate for parabolic equations with Neumann boundary conditions, enabling new observability results from measurable time sets in bounded domains.

Contribution

It introduces a novel global estimate for solutions of parabolic equations with Neumann conditions, extending unique continuation principles with quantitative bounds.

Findings

Established a global quantitative estimate of unique continuation.

Derived an observability inequality from measurable sets in time.

Utilized Carleman commutator estimates and frequency function methods.

Abstract

In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.