Observation estimate for the heat equations with Neumann boundary condition via logarithmic convexity

TL;DR

This paper establishes a unique continuation inequality for heat equations with Neumann boundary conditions using a global approach and logarithmic convexity, advancing understanding of heat propagation and control.

Contribution

It introduces a novel method combining Carleman estimates and frequency function analysis to prove unique continuation for heat equations with Neumann boundary conditions.

Findings

Proves a Hölder-type inequality for heat equations with potential and Neumann boundary conditions.

Develops a refined parabolic frequency function method for global analysis.

Achieves a new logarithmic convexity property for the frequency function.

Abstract

We prove an inequality of H\"older type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.