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

变分法的核心问题是在给定的空间上全局或局部地最小化给定的函数（通常称为泛函或能量）。这可以从找到连接给定表面上的两个点的最短路径的问题到找到具有最小总能量的液晶中的指向矢场的问题。当一个人试图系统地研究这类问题时，不仅要找到极小化器（如果存在的话），而且要研究这类极小化器的全集及其关键性质，例如，它有多大：有限还是无限？它是否包含某些对称性？正是在解决这类问题时，人们立即被引导去研究能量和底层空间在全球范围内相互作用的方式和形式（这就引入了拓扑学的数学概念）。一类特殊的问题，提出的研究直接涉及到出现在非线性弹性理论。在这里，弹性材料在受到外部激励[施加的力或边界位移]时的响应是通过最小化弹性能量来描述的，该弹性能量定义在所有可能变形的无限维空间上。平衡态则对应于各种类型的极小值（在适当的范数下是全局的或局部的）或极值（它们仅仅使能量沿沿着所有假设的变化保持平稳）。弹性意味着能量泛函不直接依赖于变形本身，而是依赖于变形梯度（在每个空间点处是3 × 3矩阵）。材料的特定选择（例如，金属对橡胶）仅通过规定和影响存储能量密度（其是后一矩阵空间上的函数）的选择的本构假设而进入。为了进行成功的建模和分析，非常重要的是，所存储的能量密度的性质反映并与物理学完全一致，而不是为了方便和简单的数学而简化。如果忽视这一点，就会对平衡态的多重性、稳定性的交换等问题的研究产生严重的后果。（例如，在屈曲和滞后问题中），动态稳定性，奇异性的形成和性质（例如断裂和气蚀），结果表明，这些存储能量密度应该落入的一般框架是由Morrey发现并引入的准凸性通过在变分学中研究适当弱拓扑中的下连续性这一明显独立的途径，不幸的是，尽管在过去的60年里，在变分法中进行了深入的研究，并在大约40年前发现了弹性材料的拟凸性和本构假设之间的紧密关系，但对拟凸性的理解仍然很差，我们知道的这种函数的真正例子很少。这种情况部分是由于拟凸性定义的特殊方式，部分是由于没有有效的方法来决定一个给定的函数是否是拟凸的。因此，可以公平地说，这样的拟凸性确实仍然是一个神秘的性质！本研究的目的是进一步研究这一概念，并解决一些悬而未决的问题，在其心脏。这将与系统研究的基本空间的拓扑结构的方向保持和体积保持的地图，是巨大的重要性，不仅在弹性理论，但在功能理论，几何和分析相结合。预计这项调查的结果将导致设计新的方法和技术在处理问题的拟凸性，正则性理论和拓扑结构，并将开辟新的前沿课题。在更大的范围内，这将是应用数学家，材料科学家和生物学家的极大兴趣。

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.