Extension Operators for Sobolev Spaces on Periodic Domains, Their Applications, and Homogenization of a Phase Field Model for Phase Transitions in Porous Media

Abstract

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.