Carleman estimates for higher order partial differential operators and its applications

TL;DR

This paper introduces a new back-propagation method to derive Carleman estimates for higher order partial differential operators, enabling applications like proving stability in fractional diffusion equations.

Contribution

A novel back-propagation method for establishing Carleman estimates for complex differential operators, with potential for broader applications and numerical analysis.

Findings

Developed a new back-propagation method for Carleman estimates

Proved conditional stability of a Cauchy problem for fractional diffusion

Method applicable to other PDE operators

Abstract

In this paper, we obtain a Carleman estimate for the higher order partial differential operator. In the process of establishing this estimate, we developed a new method, which is called the back-propagation method (the BPM, for short). This method can also be used to build up Carleman estimates for some other partial differential operators, and might provide assistance with corresponding numerical analyses. As an application of the above-mentioned Carleman estimate, we proved the conditional stability of a Cauchy problem for a time fractional diffusion equation.