Asymptotic analysis of the diffusion-absorption equation with fast and strongly oscillating absorbtion coefficient in the two-dimensional case

The Helmholtz equation with fast oscillating absorbtion coefficient and constant reflection coefficient is considered. The equation models light absorption in a medium containing a periodic set of fine blood vessels. It is assumed that the absorption takes place only inside the vessels. It is also assumed that the reflection coefficient is constant whereas the absorbtion coefficient is small everywhere except for a set of periodic thin strips modeling blood vessels, where the absorption coefficient equals a large parameter $\omega$. There are two other parameters in the problem: $\varepsilon$ is the ratio of the distance between the vessel axes to a characteristic macroscopic size, and $\delta$ is the ratio of the width of the fine vessels to the period. Both parameters $\varepsilon$ and $\delta$ are assumed to be small. The main result is the construction of an asymptotic solution.