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**Estimates of the integral kernels arising from inverse problems for a three-dimensional heat equation in thermal imaging.**
*(English)*
Zbl 1288.31006

Summary: This paper studies precise estimates of integral kernels of some integral operators on the boundary \(\partial D\) of bounded and strictly convex domains with sufficiently regular boundary. Assume that an integral operator \(K_\mu\) on \(\partial D\) has the integral kernel \(K_\mu(x,y)\) with estimate

\[ | K_\mu(x,y)|\leq C\mu e^{-\mu| x-y|},\qquad x,y\in\partial D,\quad\mu\gg 1. \]

Then, from the Neumann series, the operator \(K_\mu(I-K_\mu)^{-1}\) is also an integral operator. The problem is whether the integral kernel of \(K_\mu(I-K_\mu)^{-1}\) can be estimated by the term \(\mu e^{-\mu| x-y|}\) up to a constant or not. If the boundary \(\partial D\) is strictly convex, such types of estimates hold.

The most important point is that the obtained estimates have the same decaying behavior as \(\mu\to\infty\) and the same exponential term as for the original kernel \(K_\mu(x,y)\). These advantages are essentially needed to handle some inverse initial boundary value problems whose governing equation is the heat equation in three dimensions.

\[ | K_\mu(x,y)|\leq C\mu e^{-\mu| x-y|},\qquad x,y\in\partial D,\quad\mu\gg 1. \]

Then, from the Neumann series, the operator \(K_\mu(I-K_\mu)^{-1}\) is also an integral operator. The problem is whether the integral kernel of \(K_\mu(I-K_\mu)^{-1}\) can be estimated by the term \(\mu e^{-\mu| x-y|}\) up to a constant or not. If the boundary \(\partial D\) is strictly convex, such types of estimates hold.

The most important point is that the obtained estimates have the same decaying behavior as \(\mu\to\infty\) and the same exponential term as for the original kernel \(K_\mu(x,y)\). These advantages are essentially needed to handle some inverse initial boundary value problems whose governing equation is the heat equation in three dimensions.

### MSC:

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

35R30 | Inverse problems for PDEs |

35K05 | Heat equation |

### Keywords:

integral operators; bounded strictly convex domains; inverse initial boundary value problems### References:

[1] | M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method , submitted. · Zbl 1328.35310 |

[2] | M. Ikehata and M. Kawashita, Asymptotic behavior of the solutions for Laplacian with the inhomogeneous Robin type conditions in bounded domains with the large parameter , in preparation. · Zbl 1403.35191 |

[3] | S. Mizohata, Theory of Partial Differential Equations , trans. K. Miyahara, Cambridge Univ. Press, New York, 1973. · Zbl 0263.35001 |

[4] | S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients , Comm. Pure. Appl. Math. 20 (1967), 431-455. · Zbl 0155.16503 · doi:10.1002/cpa.3160200404 |

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