Unbounded Functionals in the Calculus of Variations. Representation, Relaxation, and Homogenization
Mathematical modelling of stationary problems often leads to minimising energy functionals of the type F(Ω, u) = ∫Ω f(x, ∂u) dx. In standard situations, the functional is locally bounded and lower semicontinuous in some Sobolev space, or in the space of functions of bounded variation. However, a more careful analysis allows one to consider functionals for which the reasonable function spaces remain the same, but the functional can be unbounded on bounded subsets of the space, or attain infinite values. In this book, the authors are primarily motivated by examples in which even the energy density function f may be somewhere infinite: this appears as a natural way of handling the presence of a constraint.
To keep the volume as self-contained as possible, five chapters of preliminaries are included. Here the reader can find the foundations of measure and integration theory and an introduction to some function spaces and variational methods in a very abstract setting. Chapter 6 surveys some classical results for finite-valued functionals, as a counterpart to later results in the ‘unbounded’ theory, and three physically motivated examples are formulated: elastoplastic torsion problems, modelling of non-linear elastomers and electrostatic screening. Chapter 7 starts a systematic treatment of ‘unbounded’ functionals. Abstract regularisation and Jensen’s inequality yield a very general result on lower semicontinuity of convex functionals. The unique extension problem, like the relaxation problem, asks for an extension of a functional which is well defined for smooth functions but lacks meaning for more general functions: its analysis is performed in an axiomatic setting. The problem of integral representation asks which functionals on a given function space can be represented as integrals in the form shown above (or, in a more general form, with a part singular with respect to the Lebesgue measure). The general integral representation theory helps to answer such questions for functionals arising as relaxed functionals or Г-limits. The relaxation problem is studied for Neumann and Dirichlet boundary data and the answer is given with full identification of the integrand. In Chapters 11-14, the homogenisation theory for unbounded functionals is developed: this studies the limit behaviour of fine periodical structures as the period tends to 0, and simulates a macroscopic view of microstructured materials. In this book, the main feature of the analysis is ‘unboundedness’ of the functional. The limit process is described by the Г-convergence and applied to convergence of minima and minimisers. The homogenisation procedure is investigated in the general case, then in case of special constraints, and finally for the particular models arising from physics.
This book contains old and new results from a significant part of the calculus of variations. The authors are well-known experts in the field whose approach is unified and elegant. The book is primarily aimed at graduate students and researchers in mathematics, but may be useful and comprehensive for a broader community of applied mathematicians, physicists and engineers.