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For a fully-coupled Darcy-Stokes system describing the exchange of fluid and stress balance across the interface between a saturated porous medium and an open very narrow channel, the limiting problem is characterized as the width of the channel converges to zero. It is proven that the limit problem is a fully-coupled system of Darcy flow in the porous medium with Brinkman flow in tangential coordinates of the lower dimensional interface., Comment: 25 pages, 4 figures

The first part of this thesis is concerned with extension operators for Sobolev spaces on periodic domains and their applications. When homogenizing nonlinear partial differential equations in periodic domains by two-scale convergence, the need for uniformly bounded families of extension operators often arises. In this thesis, new extension operators that allow for estimates in the whole domain, even if the complement of the periodic domain is connected, are constructed. These extension operators exist if the domain is generalized rectangular. They are useful for homogenization problems with flux boundary conditions. Additionally, the existence of extension operators that respect zero and nonnegative traces on the exterior boundary is shown. These can be applied to problems with mixed or Dirichlet boundary conditions. Making use of this type of extension operators, uniform Poincare and Korn inequalities for functions with mixed boundary values in periodic domains are proven. Furthermore, a generalization of a compactness theorem of Meirmanov and Zimin, which is similar to the well-known Lions-Aubin compactness theorem and which is applicable in periodic domains, is presented. The above results are then applied to the homogenization by two-scale convergence of some quasilinear partial differential equations and variational inequalities with operators of Leray-Lions type in periodic domains with mixed boundary conditions. In the second part, a phase field model for phase transitions on the pore scale of a porous medium is introduced. The existence and uniqueness of weak solutions is shown. Uniform a-priori estimates are established by making use of extension operators of the type of those that have previously been constructed in the first part of this thesis. By using two-scale convergence and again applying the extension operators, a homogenized phase field model is obtained. Finally, by applying the method of formal asymptotic expansion, a macroscopic sharp interface model for phase transitions in porous media is derived. Possible applications include the melting of permafrost soil and the frost attack on concrete.

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.