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 $\partial D$ 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 $\left|K_{\mu }\right(x,y\left)\right|\le C\mu e^{-\mu |x-y|}$ ($x,y\in \partial D$, $\mu \gg 1$). 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.