A shape optimization approach for an inverse heat source problem (Q2918851)

This paper investigates the inverse heat source problem of the determination of heat conduction properties of the medium from additional information about the heat flux. More precisely, the authors consider the inverse problem in which the boundary of a sample is accessible to temperature and flux measurements while a subdomain is inaccessible, and is to be determined from additional Neumann data on a part of the boundary of the sample.NEWLINENEWLINESuch kinds of problems appear in the thermal imaging technique which consists of applying heat flux to the surface of the sample to be imaged and then monitoring the surface temperature response of the sample over time. From flux-temperature measurements, it is possible to estimate certain internal properties of the sample or to determine the shape of some inaccessible subdomains of the sample.NEWLINENEWLINESo far, several papers were published devoted to the study of uniqueness and stability results for this inverse problem in the steady state case. However, the parabolic cases were insufficiently considered. The main purpose of the paper is to provide a generalization of \textit{M. Choulli}'s stability result [Inverse Probl. 19, No. 4, 895--907 (2003; Zbl 1054.35124); Appl. Anal. 85, No. 6--7, 693--699 (2006; Zbl 1101.35078)] to the parabolic equation by adapting the same strategies to the considered problem and using appropriate functional spaces. It should be noted that, for this study, it was necessary to overcome some technical difficulties due to time dependence.NEWLINENEWLINEThe main result of the paper consists in establishing a local stability estimate for the problem of determining a domain which appears in the heat equation from additional Neumann data on a part of the boundary. This estimate is obtained by taking advantage of some shape optimization tools.