Mathematical Sciences: Boundary Value Problems, Unique Continuation and Schrodinger Operators

沈DMS-9500635 这个项目继续数学研究的问题， 偏微分方程将在边界值上进行工作 具有粗糙边界的区域中的椭圆型方程组的问题， 在实践中自然产生。更准确地说，预解估计和 三维Stokes算子的分数幂 将研究Lipschitz域。所获得的结果将用于 在研究模拟流动的Navier-Stokes方程时， 一种粘性不可压缩的流体。采用的方法包括 奇异积分，层势，通过 部分积分和插值。第二条研究路线 涉及具有退化电势的薛定谔算子 和磁场，控制着粒子的动力学， 量子力学工作将在一定的有界性 类似于Riesz变换的算子， 的非经典本征值渐近性和指数衰减 本征函数将采用的方法基于一个改进的版本， 不确定性原则最后，我们将继续研究 退化椭圆算子的延拓问题 偏微分方程是数学的主干 物理学中的建模数学分析的作用 是提供有关的定性和定量信息 解决方案这包括对存在问题的回答， 唯一性和平滑性。该奖项支持的研究， 除了对偏微分方程感兴趣之外， 在应用数学领域非常重要， 数学物理 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Shen DMS-9500635 This project continues mathematical research on problems in partial differential equations. Work will be done on boundary value problems for elliptic systems in domains with rough boundaries which arise naturally in practise. More precisely, resolvent estimates and fractional powers of the Stokes operator in three-dimensional Lipschitz domains will be studied. The results obtained will be used in the investigation of Navier-Stokes equations which model the flow of a viscous incompressible fluid. Methods to be employed include singular integrals, layer potentials, formulas obtained through integration by parts, and interpolation. A second line of research concerns Schrodinger operators with degenerate electrical potentials and magnetic fields, which governs the dynamics of particles in quantum mechanics. Work will be done on boundedness of certain operators which are the analogues of Riesz Transforms, the non-classical eigenvalue asymptotics and exponential decays of eigenfunctions. The approach to be used is based on a refined version of uncertainty principal. Finally work will continue on the unique continuation problem for the degenerate elliptic operators. Partial differential equations form the backbone of mathematical modeling in the physical science. The role of mathematical analysis is to provide qualitative and quantitative information about the solutions. This include answers to questions about existence, uniqueness and smoothness. The research supported by this award, aside from its interest in partial differential equations, is very important in the areas of applied mathematics and mathematical physics. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%