Green functions and propagation of waves in strongly inhomogeneous media

TL;DR

This paper investigates how Green functions behave anomalously in media with highly variable properties, affecting wave and diffusion propagation in such inhomogeneous environments.

Contribution

It reveals the anomalous behavior of Green functions for differential operators with singular or unbounded coefficients and explores implications for wave and diffusion propagation.

Findings

Green functions can exhibit unexpected short or long-distance behavior.

Anomalous Green function behavior impacts wave and diffusion dynamics.

Results highlight the importance of media inhomogeneity on differential operator solutions.

Abstract

We show that Green functions of second-order differential operators with singular or unbounded coefficients can have an anomalous behaviour in comparison to the well-known properties of Green functions of operators with bounded coefficients. We discuss some consequences of such an anomalous short or long distance behaviour for a diffusion and wave propagation in an inhomogeneous medium.