# American Institute of Mathematical Sciences

November  2014, 8(4): 1073-1116. doi: 10.3934/ipi.2014.8.1073

## An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method

 1 Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashi Hiroshima 739-8527, Japan 2 Department of Mathematics, Graduate School of Sciences, Hiroshima University, Higashi Hiroshima 739-8526, Japan

Received  December 2013 Revised  October 2014 Published  November 2014

This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A single set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.
Citation: Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems and Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073
##### References:
 [1] K. Bryan and F. L. Caudill, Jr., Uniqueness for a boundary identification problem in thermal imaging, in Differential Equations and Computational Simulations III (eds. J. Graef, R. Shivaji, B. Soni and J. Zhu ), Electronic Journal of Differential Equations, Conference 01 (1998), 23-39. Available from: http://www.ma.hw.ac.uk/EJDE/index.html. [2] A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (eds. W. H. Meyer and M. A. Raupp), Brazilian Math. Society, Rio de Janeiro, (1980), 65-73. [3] B. Canuto, E. Rosset and S. Vessella, Quantitative estimate of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries, Trans. Amer. Math. Soc., 354 (2002), 491-535. doi: 10.1090/S0002-9947-01-02860-4. [4] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for sciences and technology, Evolution problems I, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [5] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308. [6] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. doi: 10.1515/jiip.2000.8.4.367. [7] M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005. doi: 10.1080/00036810701460834. [8] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010, 20pp. doi: 10.1088/0266-5611/26/5/055010. [9] M. Ikehata, The framework of the enclosure method with dynamical data and its applications, Inverse Problems, 27 (2011), 065005, 16pp. doi: 10.1088/0266-5611/27/6/065005. [10] M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005, 10pp. doi: 10.1088/0266-5611/25/7/075005. [11] M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems, 26 (2010), 095004, 15pp. doi: 10.1088/0266-5611/26/9/095004. [12] M. Ikehata and M. Kawashita, Estimates of the integral kernels arising from inverse problems for a three-dimensional heat equation in thermal imaging, Kyoto J. Math., 54 (2014), 1-50. doi: 10.1215/21562261-2400265. [13] S. Mizohata, Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973. [14] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [15] S. Vessella, Stability estimates in an inverse problem for a three-dimensional heat equation, SIAM J. Math. Anal., 28 (1997), 1354-1370. doi: 10.1137/S0036141095294262. [16] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Topical Review, Inverse Problems, 24 (2008), 023001, 81 pp. doi: 10.1088/0266-5611/24/2/023001.

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##### References:
 [1] K. Bryan and F. L. Caudill, Jr., Uniqueness for a boundary identification problem in thermal imaging, in Differential Equations and Computational Simulations III (eds. J. Graef, R. Shivaji, B. Soni and J. Zhu ), Electronic Journal of Differential Equations, Conference 01 (1998), 23-39. Available from: http://www.ma.hw.ac.uk/EJDE/index.html. [2] A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (eds. W. H. Meyer and M. A. Raupp), Brazilian Math. Society, Rio de Janeiro, (1980), 65-73. [3] B. Canuto, E. Rosset and S. Vessella, Quantitative estimate of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries, Trans. Amer. Math. Soc., 354 (2002), 491-535. doi: 10.1090/S0002-9947-01-02860-4. [4] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for sciences and technology, Evolution problems I, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [5] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308. [6] M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. doi: 10.1515/jiip.2000.8.4.367. [7] M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Applicable Analysis, 86 (2007), 963-1005. doi: 10.1080/00036810701460834. [8] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010, 20pp. doi: 10.1088/0266-5611/26/5/055010. [9] M. Ikehata, The framework of the enclosure method with dynamical data and its applications, Inverse Problems, 27 (2011), 065005, 16pp. doi: 10.1088/0266-5611/27/6/065005. [10] M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005, 10pp. doi: 10.1088/0266-5611/25/7/075005. [11] M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems, 26 (2010), 095004, 15pp. doi: 10.1088/0266-5611/26/9/095004. [12] M. Ikehata and M. Kawashita, Estimates of the integral kernels arising from inverse problems for a three-dimensional heat equation in thermal imaging, Kyoto J. Math., 54 (2014), 1-50. doi: 10.1215/21562261-2400265. [13] S. Mizohata, Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973. [14] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291. [15] S. Vessella, Stability estimates in an inverse problem for a three-dimensional heat equation, SIAM J. Math. Anal., 28 (1997), 1354-1370. doi: 10.1137/S0036141095294262. [16] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Topical Review, Inverse Problems, 24 (2008), 023001, 81 pp. doi: 10.1088/0266-5611/24/2/023001.
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