Variational convergence of nonlinear partial differential operators on varying Banach spaces

Research output: ThesisDoctoral ThesisMonograph

Abstract

In this doctoral thesis, a new approach towards variational convergence of quasi-linear monotone partial differential operators is elaborated. To this end, we analyze more explicitly the so-called Kuwae-Shioya convergence of metric spaces in the case of Banach spaces. For the first time, weak Banach space topologies are included. We achieve the objective to be able to formulate reasonable (topological) statements about convergence of vectors, functionals or operators such that each element of a convergent sequence lives in/on another distinct Banach space. Banach space convergence is considered a natural generalization of Gromov-Hausdorff convergence of compact metric spaces. The associated theory is developed here completely and justified by several examples. Among other things, we are able to consider varying Lpn(Omegan,mathcalFn,mun) spaces such that the measurable space (Omegan,mathcalFn) as well as the measure mun as well as the degree of integrability pn varies for positive integers n, and such that the limit ntoinfty is given sense. Inside the framework of varying spaces, we show that a number of classical results on variational convergence still hold. Explicit applications are given for the equivalence of so-called Mosco convergence of convex functionals and the strong graph convergence of the associated subdifferential operators. In the case of abstract Lp-spaces, we prove an elaborate result yielding a general transfer method that enables us to carry over classical results (for one fixed space) to the case of varying spaces. More precisely, we construct isometries that respect the asymptotic topology of the varying Banach spaces and allow us to transform back to one fixed Banach space. We are considering four types of quasi-linear partial differential operators mapping a Banach space X to its dual space Xast. All of these operators are characterized completely via variational methods by lower semi-continuous convex functionals on a Banach space V embedded properly into X. As operators to be approximated, we present the weighted (non-homogeneous) Phi-Laplacian in mathbbmRd, the weighted p-Laplacian in mathbbmRd, the 1-Laplacian with vanishing trace in a bounded domain, and the generalized porous medium resp. fast diffusion operator in an abstract measure space. When taking the Mosco approximation of their energies, we generally vary the weights (measures). In the second and third case, p is also varied. When dealing with approximations of such kind, varying spaces occur naturally. In the theory of homogenization, the special case of two-scale convergence has already been being employed for some time. Furthermore, we develop an alternative approach towards weighted p-Sobolev spaces of first order, which enables us to consider weights in a class different from the Muckenhoupt class. We prove a new result on density of smooth functions in weighted p-Sobolev spaces, which is known and well-studied as "Markov uniqueness" for p=2. This problem is also known as "H=W", that is, the coincidence of the strong and the weak Sobolev space. With the help of this result, we are able to identify the Mosco limit of weighted p-Laplace operators.
Original languageEnglish
Place of PublicationBielefeld, Germany
Publisher
Publication statusPublished - 2010
MoE publication typeG4 Doctoral dissertation (monograph)

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