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

摘要费尔德曼该项目致力于研究具有约束、移动和自由边界的变分问题。研究的第一个领域是研究与Monge-Kantorovich质量传递问题相关的具有逐点梯度约束的演化问题。一个这样的演化问题是作为坍塌沙堆的模型出现的，还有其他例子，包括压缩模塑和第二类超导模型。在作者与L.C.Evans和R.F.Gariepy的合作中，坍塌沙堆模型与一个几何演化问题有关，其中每个点上移动表面的速度由表面的局部和非局部几何决定。这个几何问题具有非局部性质的“抛物性”性质。该研究的主要思想是构造几何演化问题的粘性解，研究这些解的正则性和渐近性，并研究几何演化问题与原变分演化问题之间的联系。研究的第二个领域是研究另一类带约束的变分发展问题--调和映射的热流。在以前的工作中，作者证明了热流到满足某种稳定性条件的球体中的部分正则性--一种熵型变分条件。接下来的步骤包括研究这种热流的存在和唯一性，以及研究热流进入球面以外的紧致流形。第三个研究领域是将L.Caffarelli关于线性方程两相问题自由边界正则性的一些著名结果推广到某些类型的非线性椭圆型方程。在前人的工作中，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，即当解满足由自由边界分隔的两个子区域中的两个不同的线性椭圆型方程时。科学和工程中的许多问题自然而然地导致数学模型具有带约束的变分演化问题的形式。其中一类问题是逐点梯度约束的演化问题，包括坍塌沙堆模型、压缩成型模型和第二类超导模型。演化问题解的定性性质本身和可能的应用都是令人感兴趣的。具有逐点梯度约束的变分发展问题的典型性质包括形成移动的自由边界。所提出的研究的第一个领域是通过将这些移动边界与移动曲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移动边界。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边界的正则性(光滑性)。另一类带约束的变分发展问题是调和映射的热流问题。这样的问题出现在几何学中，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些类调和映射的热流的存在唯一性。

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.