Quantification of the unique continuation property for the heat equation

,Laurent Bourgeois

doi:10.3934/mcrf.2017012

Abstract

In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method.

Highlights

This paper deals with the ill-posed problem of finding the solution to the heat equation from a source term and lateral Cauchy data
Problems such as (1) arise in the framework of inverse problems related to the heat equation, where f = 0 and the data (g0, g1) correspond to measurements on the accessible part Γ0 of the boundary of Ω
We show that the Main Theorem enables us to estimate the initial condition and to obtain a convergence rate for the method of quasi-reversibility

Summary

Laurent Bourgeois

Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, AIMS, 2017, 7 (3), pp.347 - 367. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Manuscript submitted to AIMS’ Journals Volume X, Number 0X, XX 200X doi:10.3934/xx.xx.xx.xx pp.

Introduction

Given that

By gathering the above estimates we obtain that

We can choose χε such that

Using the fact that