Mathematical Sciences: Mass Transfer, Heat Flows with Constraints, Moving and Free Boundaries

数学科学：传质、约束热流、移动边界和自由边界

• 批准号: 9623276
• 负责人: Mikhail Feldman
• 金额: $ 7.97万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623276&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Mass; Transfer; Heat

Abstract Feldman The project is devoted to the study of variational problems with constraints, moving and free boundaries. The first area of the research is the study of evolution problems with pointwise gradient constraints, related to Monge-Kantorovich mass transfer problem. One such evolution problem arises as a model of collapsing sandpiles, and there are other examples, including models of compression molding and type II superconductivity. In the joint work of the proposer with L.C.Evans and R.F.Gariepy, the collapsing sandpiles model was related to a geometric evolution problem where the velocity of a moving surface at each point is determined by both local and nonlocal geometry of the surface. This geometric problem possesses some "parabolicity" properties of nonlocal nature. The main idea of the proposed research is to construct viscosity solutions of the geometric evolution problem, to study regularity and asymptotic properties of these solutions, and to study connections of the geometric evolution problem with the original variational evolution problem. The second area of the research is the study of another class of variational evolution problems with constraints - heat flows for harmonic maps. In the previous work the proposer has proved partial regularity for heat flows into spheres satisfying some stability condition - a variational condition of entropy type. The next steps include the study of existence and uniqueness of such heat flows, and the study of heat flows into compact manifolds other then sphere. The third area of research is to try to extend some of celebrated results of L.Caffarelli on regularity of free boundaries in two-phase problems for linear equations to some classes of nonlinear elliptic equations. In previous work the proposer proved regularity for anisotropic two-phase problem with Lipschitz free boundary, i.e., in the case when the solutions satisfy two different linear elliptic equations in two subregions separated by free boundary. Many problems in science and engineering lead naturally to mathematical models that have the form of variational evolution problems with constraints. One class of such problems, the evolution problems with pointwise gradient constraints, includes models of collapsing sandpiles, of compression molding, of type II superconductivity. Qualitative properties of solutions of evolution problems are of interest by itself and with respect to possible applications. Typical properties of variational evolution problems with pointwise gradient constraints include formation of moving free boundaries. The first area of the proposed research is the study of these moving boundaries by relating them to a geometric evolution problem for a moving surface and studying solutions of this geometric problem. Free boundary problems arise also in the static case. In this case the free boundaries can be viewed as a model of the interface between two different substances. The second area of the research is the study of regularity (smoothness) properties of such boundaries for some nonlinear static problems. Another class of variational evolution problems with constraints is heat flows for harmonic maps. Such problems arise in geometry and their solutions have interesting properties. These heat flows represent one of the models of liquid crystals. The third area of the research is the study of existence and uniqueness of heat flows for harmonic maps of certain classes.

该项目致力于研究具有约束、移动边界和自由边界的变分问题。研究的第一个领域是研究具有点向梯度约束的演化问题，与Monge-Kantorovich传质问题有关。一个这样的演化问题出现在崩塌砂堆模型中，还有其他的例子，包括压缩成型模型和II型超导模型。在与L.C.Evans和R.F.Gariepy的联合工作中，将崩塌沙堆模型与一个几何演化问题联系起来，其中运动表面在每一点的速度由表面的局部和非局部几何形状决定。这个几何问题具有一些非局部的“抛物线”性质。本研究的主要思想是构造几何演化问题的黏性解，研究这些解的正则性和渐近性质，并研究几何演化问题与原变分演化问题的联系。研究的第二个领域是研究另一类带约束的变分演化问题-调和映射的热流。在之前的工作中，作者证明了热流在满足某些稳定性条件——熵型变分条件下的部分规律性。下一步包括对这种热流的存在性和唯一性的研究，以及对非球面紧致流形的热流的研究。第三个研究领域是尝试将L.Caffarelli关于线性方程两相问题中自由边界正则性的一些著名结果推广到某些非线性椭圆型方程。在以往的工作中，作者证明了具有Lipschitz自由边界的各向异性两相问题的正则性，即当解满足两个不同的线性椭圆方程时，在两个被自由边界分隔的子区域内。科学和工程中的许多问题自然会导致具有带约束的变分进化问题形式的数学模型。这类问题的一类，具有点向梯度约束的演化问题，包括崩塌沙堆、压缩成型和II型超导的模型。演化问题解的定性性质本身及其可能的应用都是令人感兴趣的。具有点向梯度约束的变分演化问题的典型性质包括运动自由边界的形成。研究的第一个领域是通过将这些运动边界与运动表面的几何演化问题联系起来，并研究该几何问题的解来研究这些运动边界。在静态情况下也会出现自由边界问题。在这种情况下，自由边界可以看作是两种不同物质之间界面的模型。研究的第二个领域是研究一些非线性静态问题的边界的正则性（平滑性）。另一类带约束的变分演化问题是调和映射的热流问题。这类问题出现在几何中，它们的解具有有趣的性质。这些热流代表了液晶的一种模型。研究的第三个领域是研究某类调和图的热流的存在唯一性。