We establish a probe type reconstruction scheme for identifying an inclusion inside a heat conductive medium by nondestructive testing called thermography. For the one space dimension, this has been already achieved by Y. Daido, H. Kang and G. Nakamura. The present paper shows that their result can be generalized to higher space dimension.

@article{ASNSP_2010_5_9_4_725_0, author = {Isakov, Victor and Kim, Kyoungsun and Nakamura, Gen}, title = {Reconstruction of an unknown inclusion by thermography}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {4}, year = {2010}, pages = {725-758}, zbl = {1215.35169}, mrnumber = {2789473}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0} }

Isakov, Victor; Kim, Kyoungsun; Nakamura, Gen. Reconstruction of an unknown inclusion by thermography. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 4, pp. 725-758. http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0/

[1] Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal. 37 (2005), 200–217. | MR 2176930 | Zbl 1099.35162

and ,[2] Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 22 (1968), 607–694. | Numdam | MR 435594 | Zbl 0182.13802

,[3] The significance of damage and defects and their detection in composite materials: A review, J. Strain Anal. Eng. Des. 27 (1992), 29–42.

and ,[4] A probe method for the inverse boundary value problem of non-stationary heat equations, Inverse Problems 23 (2007), 1787–1800. | MR 2353315 | Zbl 1126.35092

, and ,[5] Estimate of the fundamental solution for parabolic operators with discontinuous coefficients, preprint (http://arxiv.org).

, and ,[6] Stable determination of the discontinuous conductivity coefficient of a parabolic equation, preprint (http://arxiv.org). | MR 2596551 | Zbl 1222.35210

and ,[7] On uniqueness of recovery of the discontinuous conductivity coefficient of a parabolic equation, SIAM J. Math. Anal. 28 (1997), 49–59. | MR 1427727 | Zbl 0870.35124

and ,[8] “Partial Differential Equations of Parabolic Type", Prentice-Hall, INC. 1964. | MR 181836 | Zbl 0144.34903

,[9] An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163–218. | MR 331441 | Zbl 0238.35038

,[10] “The Analysis of Linear Partial Differential Operator III", A series of comprehensive study in Mathematics, Springer-Verlag, 1985. | MR 781536

,[11] Reconstruction of inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 10 (2002) 37–65. | MR 1889237 | Zbl 0994.35120

,[12] “Inverse Problems for Partial Differential Equations", Applied Mathematical Sciences, Vol. 127, Springer 1988.

,[13] “Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity", North-Holland series in Applied Mathematics and Mechanics, Vol. 25, North-Holland Publishing Company, 1979. | MR 530377

,[14] “Second Order Parabolic Differential Equations", World Scientific Publishing Co., Inc., 1996. | MR 1465184 | Zbl 0884.35001

,[15] Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892–925. | MR 1990481 | Zbl 1125.35339

and ,[16] A review of image analysis techniques applied in transient thermographic nondestructive testing, Nondestruct. Test. Eval. 6 (1992) 343–364.

, and ,[17] Inverse method using infrared thermography for surface temperature and heat flux measurements, 20th International Congress on Instrumentation in Aerospace Simulation Facilities, 2003, ICIASF ’03, 118–126.

, and ,[18] Numerical implementation of dynamical probe method, J. Comput. Math. 28 (2010), 87–104. | MR 2603583 | Zbl 1224.65220

, and ,[19] “Partial Differential Equations", Cambridge University Press, 1987. | MR 895589 | Zbl 0623.35006

,