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

小行星9500940 这是一份向美国国家科学基金会提交的关于非线性部分 微分方程到相变、演化曲面和随机控制。在演化相变和演化曲面方面的研究涉及到反应扩散方程系统渐近分析的发展（包括相场、Cahn-Hilliard方程和向量值Ginzburg-Landau方程），进一步研究三相或多相界面的传播，分析任意余维表面的动力学，以及这些表面的水平集方法的发展。在随机控制的建议研究关注的数学金融模型的分析和近似的复杂，控制系统的简单和更容易处理的连续模型。 粘性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。 这里提出的是继续这方面的研究时，连续模型的奇异控制。 在金融数学中，偏微分方程无论在分析还是数值计算方面都是一种有效的工具 计算。 期权定价就是一个很好的例子。 主要研究者建议在分析有交易费用的模型时使用粘性解理论。 %%% 对相变和表面演化的研究与材料科学中的几种模型有关，这些模型模拟了晶界、不同相之间的界面和缺陷的动力学。 理解缺陷和界面的传播不仅对于其内在的兴趣而且对于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业、通信和数理金融等领域的问题。 在数理金融学中，研究了存在交易费用的Black-Scholes型期权定价问题。 ***

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. ***