A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry

Abstract

We construct Lebesgue and Sobolev spaces of functions defined on a continuous distribution of domains {$Y_x \subset \mathbb{R}^m$ : $x \in \Omega$}. The resulting spaces can be viewed as a generalisation of the Bochner spaces $L_p(\Omega;W_q^l(Y))$ for the case that $Y$ depends on $x \in \Omega$. Furthermore, we introduce a Lebesgue space of functions defined on the boundaries {$∂Y_x : x \in \Omega$}. The latter construction relies on a uniform Lipschitz parametrisation of the above collection of boundaries, interpreted as a higher-dimensional manifold. The results are applied to prove existence, uniqueness and upper and lower bounds for a distributed-microstructure model of reactive transport in a heterogeneous porous medium.