Mathematical Sciences: On the Qualitative Properties and Classification of Solutions to Nonlinear Differential Equations

点击翻译按钮获取中文摘要

9401441 LI This award supports mathematical research into the qualitative properties of nonlinear elliptic partial differential equations and their application to problems in geometry. Work involves the study of travelling wave fronts on infinite cylinders arising from flame propagation and ignition in combustion theory and population dynamics. Previous studies showed that the solutions of the evolutions problems approach some stable travelling wave fronts with certain speeds. There are cases where only one wave front exists. In other cases there are more and part of this investigation is to find the stable wave front. Work will also be done using the method of moving planes in the study of semilinear elliptic equations with critical and sub-critical Sobolev exponents. The goal here is to classify solutions which can then be carried over to fully nonlinear equations. Continuing work on the problem of prescribing Gaussian curvature on surfaces will also be carried out. This is sometimes referred to as the Kazdan-Warner problem. One seeks necessary and sufficient conditions for the solvability of the prescribed Gauss curvature via conformal changes of metric. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***