Mass Transfer and Evolution Problems, Free Boundary Problems

DMS-9970577摘要。费尔德曼将在三个主要议题上工作。(1)研究非均匀介质中的Monge-Kantorovich质量传递问题，其中包括带有障碍物的输运流形，以及各向异性和非光滑空间中的质量传递问题，并研究相关的演化问题。近年来，许多作者对欧氏空间中的质量输运问题的研究取得了进展。非均匀介质是自然的下一步。(2)研究三维及更高维空间中包含非局部曲率运动的几何发展方程的凸粘性解的存在性。这种几何演化与Monge-Kantorovich质量传递有关。(3)自由边界问题：解的存在性和稳定性以及自由边界的规则性。最近，传质理论方法被应用于多个应用中，包括经济学、气象学、流体动力学、沙堆生长和崩塌、非牛顿流体、塑性等数学模型。自由边界和运动界面在许多模型中是自然出现的，对这类问题的数学分析可以提高我们对模型定性性质的理解，也是发展快速可靠的数值算法的必要组成部分。此外，数学研究有时会发现不同模型之间的意想不到的联系（例如，偏微分方程和运输问题），这种相互作用在应用中很有用。

DMS-9970577ABSTRACTM. Feldman will work on three main topics. (1) Study of Monge-Kantorovichmass transfer problem in nonhomogeneous media, which include transporton manifolds with obstacles, and in anisotropic and nonsmooth spaces.Also, related evolution problems will be studied. Recently a progress was made by many authors in studying mass transport in the Euclidean space. Nonhomogeneous media is a natural next step. (2) Study of existence ofconvex viscosity solutions of geometric evolution equations, including nonlocal curvature motion, in the spatial dimensions three and higher. Such geometric evolutions are related to Monge-Kantorovich mass transfer. (3) Free boundary problems: existence and stability of solutions, and regularity of free boundaries.Methods of mass transfer theory were used recently in several applications including mathematical models in economics, meteorology, fluid dynamics, sandpiles growth and collapse, non-Newtonian fluids, plasticity. Free boundaries and moving interfaces arise naturally in many models.Mathematical analysis of such problems improves our understanding of qualitative properties of models, and is a necessary ingredient in developing of fast and reliable numerical algorithms. In addition, mathematical study sometimes discovers unexpected connections between different models (for example, partial differential equations and transport problems), and such interplay is useful in applications.