Results on the diffusion equation with rough coefficients

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.