Regularity for Partial Differential Equations

本文主要研究双调和映射的正则性理论、偏微分方程解的水平曲面、几何流动问题和退化方程。建立了弱双调和映射的正则性。双调和映射有望在四种流形理论中发挥重要作用。水平面是相变理论、几何演化理论和自由边界问题研究的中心课题。在许多实际问题中，一些特殊的水平面表现为晶体的边界和冲击波的波前。我们计划发展一些新的想法，以帮助理解这些水平面的性质、稳定性和规律性。建立了晶体流动的数学存在性和晶体边界的规律性。自从牛顿发现物理第二定律以来，从流体力学、相对论到量子力学，数学物理的进步一直依赖于偏微分方程组。虽然偏微分方程式非常精确地描述了自然，但它们也面临着许多挑战。主要的困难在于它们的非线性。复杂性的另一个重要来源是某些物理量的奇异性。从物理上讲，它们来自黑洞形成之类的现象，或者其他物理现象，如爆炸、相变，甚至是超导。我们试图通过研究奇点集的结构和构形来理解这些奇点。这类问题中有一类来自晶体生长和自由边界问题。晶体边界的运动是几何流问题。在这里，晶体的边界，例如在融化冰的情况下，是温度函数的水平表面。从这个角度研究了晶体生长的整个动力学过程。

DMS-9801374 Lihe Wang This proposal is directed in the study of regularity theory of bi-harmonic maps, level surfaces of solutions of partial differential equations, problems of geometric flows and degenerate equations. Regularity of weakly bi-harmonic maps are established. Bi-harmonic maps are expected to play an important role in the theory of four manifolds. The level surfaces are the central subject of study in the theories of phase transitions, geometric evolutions and free boundary problems. Some special level surfaces appear as the boundary of crystals and the wave fronts of shock waves in many practical problems. We plan to develop some new ideas which help to understand the nature, stability and regularity of these level surfaces. The mathematical existence of flow of crystals and the regularity of the boundary of the crystals are also established. Since Newton discovered the second law of physics, progress in mathematical physics has relied on the partial differential equations, from fluid mechanics, relativity to quantum mechanics. Although partial differential equations describe nature quite precisely, they have many challenges. The major difficulty is due to their nonlinearity. Another big source of the complications comes from the singularities of some of the physical quantities. Physically these came from a phenomenon like that of the formation of a black hole or other physical phenomena such as explosion, phase transition or even super-conductivity. We try to understand these singularities by studying the structure and the configuration of the singularity set. A class of this kind of problems comes from crystal growth and free boundary problems. The motion of the boundary of the crystal is the geometric flow problem. Here the boundary of the crystal, such as in the case of melting ice, is a level surface of the temperature function. The whole dynamical process of the crystal growth is studied from This point of view.