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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
