Mathematical Sciences: Heat Flow of Harmonic Maps

点击翻译按钮获取中文摘要

The focus of this project is the mathematical analysis of solutions of partial differential equations defined on surfaces or manifolds. The work is intuitively geometric in that it seeks to determine when smooth mappings between manifolds can be deformed into harmonic maps between the same manifolds. The question can be rephrased in terms of parabolic partial differential equations where the deformation becomes the evolution of the solution in time. Considerable work has been done on problems of this nature, especially in Euclidean space. It is known that solutions need not evolve for all time, a phenomenon commonly referred to a finite-time blow up. In the present context, the known conditions which preclude blow up are related to the dimension and sectional curvatures of the target manifold. The best results are known for two dimensional manifolds. In the present work, efforts will be made to establish global existence of the heat flow of harmonic maps with boundary in higher dimensions and to understand the character of singularities when they do occur. A second objective is that of understanding the boundary regularity of the heat flow for harmonic maps in higher dimensions. Evidence suggests that the evolving solutions to the heat equation cannot first encounter singularities at the boundary. This may be difficult to show in higher dimensions, but there is reason to believe that a complete analysis is possible in two and three dimensional manifolds. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.