Mass Transfer and Evolution Problems, Free Boundary Problems

DMS-9970577 ABSTRACTM。费尔德曼将致力于三个主要主题。(1)研究非均匀介质中的Monge-Kantorovich型传质问题，包括具有障碍物的输运流形，以及各向异性和非光滑空间中的Monge-Kantorovich型传质问题。最近，许多作者在研究欧几里得空间中的质量输运方面取得了进展。非均质媒体是自然的下一步。(2)研究了三维及更高维几何发展方程，包括非局部曲率运动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，以及自由边界的正则性。近年来，传质理论的方法在经济学、气象学、流体力学、沙堆生长和崩塌、非牛顿流体、塑性力学等数学模型中得到了广泛的应用。自由边界和移动界面在许多模型中是自然出现的，对这些问题的数学分析提高了我们对模型定性性质的理解，是开发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是有用的。

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.