Results on the diffusion equation with rough coefficients

TL;DR

This paper investigates how solutions to the stationary diffusion equation behave when the diffusivity coefficient is rough or approaches infinity in parts of the medium, establishing continuity and convergence properties of the solution maps.

Contribution

It proves strong sequential continuity of solution maps for rough diffusivities in any dimension and extends the solution operator map continuously in 1D with explicit convergence estimates.

Findings

Solution maps are strongly sequentially continuous in any dimension.

In 1D, the solution operator map extends continuously with explicit convergence estimates.

The behavior of solutions with infinite diffusivity values is characterized and controlled.

Abstract

We study the behaviour of the solutions of the stationary diffusion equation as a function of a possibly rough ($L^{\infty}$-) diffusivity. This includes the boundary behaviour of the solution maps, associating to each diffusivity the solution corresponding to some fixed source function, when the diffusivity approaches infinite values in parts of the medium. In $n$-dimensions, $n \geq 1$, by assuming a weak notion of convergence on the set of diffusivities, we prove the strong sequential continuity of the solution maps. In 1-dimension, we prove a stronger result, i.e., the unique extendability of the map of solution operators, associating to each diffusivity the corresponding solution operator, to a sequentially continuous map in the operator norm on a set containing `diffusivities' assuming infinite values in parts of the medium. In this case, we also give explicit estimates on the convergence behaviour of the map.