Mathematical Sciences: Linear and Nonlinear Partial Differential Equations

9400978鞑靼人该奖项支持对偏微分方程式理论中出现的问题进行数学研究。将考虑几个项目。第一部分是无限维含无界哈密顿量的完全非线性一阶和二阶Hamilton-Jacobi方程粘性解工作的继续。目标是在早期工作的基础上考虑间断解的存在性、与最优控制或微分对策相对应的动态规划中出现的方程以及无限维中的二阶问题。其他工作将集中于研究线性偏微分方程解的唯一延拓问题，以及确定可获得此类结果的系数的最小正则性的问题。最终的目标是得到双曲边值问题生成的半群的本质谱的完整刻画及其在镇定中的应用。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

9400978 Tataru This award supports mathematical research on problems arising in the theory of partial differential equations. Several projects are to be considered. The first is a continuation of work on viscosity solutions for fully nonlinear first and second order Hamilton-Jacobi equations with unbounded Hamiltonian in infinite dimensions. The goal is to build upon the earlier work to consider issues such as the existence of discontinuous solutions, equations arising in dynamic programming corresponding to optimal control or differential games and second order problems in infinite dimensions. Other work will focus on the study of unique continuation problems for solutions to linear partial differential equations and the problem of determining the minimal regularity of the coefficients for which such results can be obtained. A final goal is that of obtaining a complete characterization of the essential spectrum for semigroups generated by hyperbolic boundary value problems with applications in stabilization. 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. ***