TOPOLOGY OF SOBOLEV SPACES AND QUASICONVEXITY: MULTIPLICITY AND SINGULARITY ANALYSIS FOR EXTREMALS AND LOCAL MINIMIZERS

The central problem in the calculus of variations is to minimize a given function (often called functional or energy) globally or locally over a given space. This can range from the problem of finding the shortest path joining two points on a given surface to the problem of finding the director field in a liquid crystal having minimum total energy. When one attempts to systematically investigate questions of this type it becomes increasingly important not only to find a minimizer [if exists] but also to study the full set of such minimizers and its key properties, e.g., how large it is: finite or infinite? Does it entail certain symmetries? It is in addressing questions of this type that one is immediately led to investiage the way and form in which the energy and the underlying space interact with one-another globally (this brings in the mathematical concept of topology). A particular class of problems that the proposed research directly relates to arise in nonlinear theory of elasticity. Here the response of an elastic material when subjected to external excitations [applied forces or boundary displacements] is described through minimization of the elastic energy which is defined over the infinte dimensional space of all possible deformations. Equilibrium states then correspond to various classes of minimizers (global or local in suitable norms) or extremals (which merely make the energy stationary along all hypothetical variations). Being elastic means that the energy functional directly depends not on the deformation itself but on the deformation gradient (that at each spatial point is a 3 X3 matrix). The particular choice of the material (e.g., metal vs. rubber) enters only through the constitutive assumptions dictating and affecting the choice of the stored energy density (that is a function on the latter space of matrices). To make a successful modelling and analysis it is very important that the properties of the stored energy density reflect and are fully aligned with physics and not simplified for the sake of convenient and easy mathematics. This when ignored will have grave consequences in the study of questions relating to multiplicity of equilibrium states, exchange of stability (e.g. in problems of buckling and hysteresis), dynamic stability, formation and nature of singularities (e.g. fracture and cavitation), etc. It turns out that the general framework for which these stored energy densities should fall into is that of quasiconvexity discovered and introduced by Morrey through the apparently independent route of studying lower semicontinuity in suitable weak topologies in calculus of variations. Unfortunately despite the intensive investigations in the past 60 years in the calculus of variations supplemented by the discovery of the tight relation between quasiconvexity and constitutive assumptions on elastic materials about 40 years ago, quasiconvexity still is poorly understood and very few genuine examples of such functions are known to us. The situation is partly due to the peculiar way in which quasiconvexity is defined and partly due to having no efficient way of deciding whether a given function is quasiconvex or not. It is thus fair to say that as such quasiconvexity truely remains a mysterious property! The purpose of this research is to investigate this notion further and address some of the open problems that lie at its heart. This will be combined with a systematic study of the topologies of the underlying spaces of orientation-preserving and volume-preserving maps that are of massive importance not only in elasticity theory but in function theory, geometry and analysis. It is expected that the results of this investigation will lead to devising new methods and techniques in handling questions on quasiconvexity, regularity theory and topology and will open new frontiers in the subject. On a larger scale this will be of great interest to applied mathematicians, material scientists, and biologists.