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交的一份关于非线性偏微分方程在相变、演化曲面和随机控制中的应用的提案。在演化相变和演化曲面方面的研究涉及到反应扩散方程（包括相场、Cahn-Hilliard和向量值Ginzburg-Landau方程）系统的渐近分析的发展，三个或更多阶段的界面传播的进一步研究，具有任意余维的表面的动力学分析，以及这些表面的水平集方法的发展。提出的随机控制研究涉及数学金融模型的分析，以及用更简单和更易于处理的连续统模型逼近复杂的受控系统。粘度解理论已经非常成功地建立了各种近似模型和原始模型之间的联系。本文建议在连续统模型被奇异控制的情况下继续研究。在数学金融中，偏微分方程不仅是分析的有效工具，也是数值计算的有效工具。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。相变和演化表面的研究涉及到材料科学中对晶界、不同相间界面和缺陷的动力学建模的几个模型。理解缺陷和接口的传播不仅对其内在的兴趣，而且对其技术的重要性都是至关重要的。随机控制的研究涉及到制造、通信和数学金融等领域的问题。在数学金融学中，研究存在交易成本的Black-Scholes型期权定价问题。＊＊＊