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

Sobolev空间拓扑和拟凸性：极值和局部极小值的多重性和奇异性分析

• 批准号: EP/V027115/1
• 负责人: Ali Taheri
• 金额: $ 55.89万
• 依托单位: University of Sussex
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV027115%2F1
• 关键词: TOPOLOGY; SOBOLEV; SPACES; QUASICONVEXITY; MULTIPLICITY

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.

变分中的中心问题是在给定空间中全局或局部最小化给定函数(通常称为泛函或能量)。这可以从寻找连接给定表面上的两个点的最短路径的问题到在具有最小总能量的液晶中寻找指向矢场的问题。当人们试图系统地研究这类问题时，不仅要找到极小化[如果存在]，而且还要研究这种极小化的全集及其关键性质，例如，它有多大：有限还是无限？它需要一定的对称性吗？正是在解决这种类型的问题时，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(这带来了拓扑学的数学概念)。这项研究直接涉及到非线性弹性理论中出现的一类特殊问题。在这里，弹性材料在受到外部激励(外力或边界位移)时的响应是通过最小化弹性能量来描述的，该能量定义在所有可能的变形的无限维空间上。然后，平衡态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(它们只是使能量沿着所有假设的变化保持不变)。弹性意味着能量泛函不直接取决于形变本身，而取决于形变梯度(即在每个空间点上是一个3x3矩阵)。材料的特定选择(例如，金属与橡胶)仅通过规定和影响存储能量密度(即，矩阵的后一空间的函数)的选择的本构假设来进入。为了成功地进行建模和分析，存储能量密度的性质必须反映并与物理完全一致，而不是为了方便和简单的数学而简化。如果忽略这一点，将在研究与平衡态的多重性、稳定性交换(例如在屈曲和滞后问题中)、动态稳定性、奇点的形成和性质(例如断裂和空化)等问题时产生严重的后果。结果表明，这些存储能量密度应该落入的一般框架是由Morrey通过研究变分中适当的弱拓扑中的下半连续的明显独立的路线而发现并引入的拟凸性。不幸的是，尽管在过去的60年里，人们对变分法进行了深入的研究，并在大约40年前发现了弹性材料上的拟凸性与本构假设之间的紧密联系，但拟凸性仍然知之甚少，我们所知的此类函数的真实例子也很少。这种情况部分是由于定义拟凸性的特殊方式，部分是因为没有有效的方法来确定给定函数是否为拟凸。因此，公平地说，这样的拟凸性确实仍然是一个神秘的性质！这项研究的目的是进一步研究这一概念，并解决其核心的一些公开问题。这将与系统研究保方向和保体积映射的底层空间的拓扑相结合，这些映射不仅在弹性理论中具有重要意义，而且在函数论、几何和分析中也具有重要意义。期望这项研究的结果将为处理拟凸性、正则性理论和拓扑学问题提供新的方法和技术，并将开辟该学科的新前沿。在更大的范围内，这将引起应用数学家、材料科学家和生物学家的极大兴趣。

项目成果

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

• DOI: 10.1016/j.na.2023.113255
• 发表时间: 2023-03
• 期刊: Nonlinear Analysis
• 作者: A. Taheri;Vahideh Vahidifar

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

• DOI: 10.1515/anona-2022-0288
• 发表时间: 2023-01
• 期刊: Advances in Nonlinear Analysis
• 影响因子: 4.2
• 作者: A. Taheri;Vahideh Vahidifar

On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint

• DOI: 10.1016/j.na.2022.112889
• 发表时间: 2022-06
• 期刊: Nonlinear Analysis
• 作者: A. Taheri;Vahideh Vahidifar