Estimates of the integral kernels arising from inverse problems for a three-dimensional heat equation in thermal imaging

Abstract

This paper studies precise estimates of integral kernels of some integral operators on the boundary ∂D of bounded and strictly convex domains with sufficiently regular boundary. Assume that an integral operator Kμ on ∂D has the integral kernel Kμ(x,y) with estimate |Kμ(x,y)|≤Cμe−μ|x−y| (x,y∈∂D, μ≫1). Then, from the Neumann series, the operator Kμ(I−Kμ)−1 is also an integral operator. The problem is whether the integral kernel of Kμ(I−Kμ)−1 can be estimated by the term μe−μ|x−y| up to a constant or not. If the boundary ∂D is strictly convex, such types of estimates hold. The most important point is that the obtained estimates have the same decaying behavior as μ→∞ and the same exponential term as for the original kernel Kμ(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.