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

9401441 LI该奖项支持对非线性椭圆型偏微分方程的定性性质及其在几何问题中的应用的数学研究。工作涉及燃烧理论和种群动力学中火焰传播和点火引起的无限圆柱体行波前的研究。以往的研究表明，演化问题的解接近于一定速度下的稳定行波前。也有只存在一个波前的情况。在其他情况下，这种研究的一部分是寻找稳定波前。在研究具有临界和亚临界Sobolev指数的半线性椭圆方程时，也将使用移动平面的方法。这里的目标是对解进行分类，然后将其转到完全非线性方程中。继续研究曲面上高斯曲率的规定问题也将进行。这有时被称为卡兹丹-华纳问题。人们通过度规的共形变化寻求规定的高斯曲率可解的充分必要条件。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***