Homogenization Techniques for Lower Dimensional Structures

Abstract

This thesis is concerned with extensions and applications of the theory of periodic unfolding in the field of (mathematical) homogenization. The first part extends the applicability of homogenization in domains with evolving microstructure to the case of evolving hypersurfaces: We consider a diffusion-reaction equation inside a perforated domain, where also surface diffusion and reaction takes place. Upon a transformation to a referential geometry, we (formally) obtain a transformed set of equations. We show that homogenization techniques can be applied to this transformed formulation. Special emphasis is placed on possible nonlinear reaction rates on the surface, a fact which requires special results for estimation and convergence results. In the limit, we obtain a macroscopic system, where each point of the domain is coupled to a system posed in the reference (micro-)geometry. Additionally, this reference geometry is evolving. In a second part, we are concerned with an extension of the notion of periodic unfolding to some Riemannian manifolds: We develop a notion of periodicity on nonflat structures in a local fashion with the help of a special atlas. If this atlas satisfies a compatibility condition, unfolding operators can be defined which operate on the manifold. We show that continuity and compactness theorems hold, generalizing the well-known results from the established theory. As an application of this newly developed results, we apply the unfolding operators to a strongly elliptic model problem. Again, we obtain a generalization of results well-known in homogenization. Moreover, we are also able to show some additional smoothness-properties of the solution of the cell problem, and we construct an equivalence relation for different atlases. With respect to this relation, the limit problem is independent of the parametrization of the manifold.