Mathematical Sciences: Nonlinear Partial Differential Equations and Their Applications to Evolving Surfaces, Phase Transitions and Stochastic Control

数学科学：非线性偏微分方程及其在演化表面、相变和随机控制中的应用

• 批准号: 9500940
• 负责人: Halil Soner
• 金额: $ 5.66万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500940&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

9500940 Soner This is a proposal to National Science Foundation to work on the applications of nonlinear partial differential equations to phase transitions, evolving surfaces and stochastic control. The proposed studies in evolutionary phase transitions and evolving surfaces concern the development of the asymptotic analysis of systems of reaction-diffusion equations (including the phase-field, Cahn-Hilliard, and the vector valued Ginzburg-Landau equations), the further study of the interface propagation with three or more phases, the analysis of the dynamics of surfaces with it arbitrary codimension, and the development of a level set approach for these surfaces. The proposed studies in stochastic control concern the analysis of mathematical financial models and the approximation of complex, controlled systems by simpler and more tractable continuum models. The theory of viscosity solutions have been very successful in establishing the connection between various approximate and the original models. Proposed here is to continue this research when the continuum models are singularly controlled. In mathematical finance, partial differential equations have been an efficient tool not only in the analysis but also in numerical computations. Option pricing provides a good example of this. The principal investigator proposes to use the theory of viscosity solutions in the analysis of models with transaction costs. %%% Research on the phase transitions and evolving surfaces is related to several models in materials science modeling the dynamics of grain boundaries, interfaces between different phases and defects. Understanding the propagation of defects and interfaces is of fundamental importance not only for its intrinsic interest but also for its technological importance. Research on stochastic control is related to problems in manufacturing, communications and mathematical finance. In mathematical finance, Black-Scholes type option pricing problems will be investigated in the presence of transaction costs. ***

9500940 Soner 这是向国家科学基金会提出的一项提案，致力于研究非线性偏微分方程在相变、演化表面和随机控制中的应用。拟议的演化相变和演化表面研究涉及反应扩散方程组（包括相场、Cahn-Hilliard 和向量值 Ginzburg-Landau 方程）渐近分析的发展、具有三个或更多相的界面传播，任意余维表面的动力学分析，以及这些表面的水平集方法的开发。随机控制方面的拟议研究涉及数学金融模型的分析以及通过更简单和更容易处理的连续模型来逼近复杂的受控系统。 粘度解理论在建立各种近似模型和原始模型之间的联系方面非常成功。 这里建议在连续介质模型受到单一控制时继续这项研究。 在数学金融中，偏微分方程不仅在分析中而且在数值计算中都是有效的工具。 期权定价就是一个很好的例子。 首席研究员建议使用粘度解理论来分析具有交易成本的模型。 %%% 对相变和演化表面的研究与材料科学中模拟晶界动力学、不同相和缺陷之间的界面的几个模型有关。 了解缺陷和界面的传播至关重要，不仅因为其内在利益，而且因为其技术重要性。 随机控制的研究与制造、通信和数学金融中的问题有关。 在数学金融中，布莱克-斯科尔斯型期权定价问题将在存在交易成本的情况下进行研究。 ***