Homogenization of Elliptic and Parabolic Partial Differential Equations

随机均匀化理论确定了一种现象的平均宏观行为，这种现象受到微观随机效应的影响。例如，人们可能感兴趣的是确定具有随机分布杂质的多孔材料的一般性质，或者预测具有随机障碍物的异质介质中的种群演化。这种现象通常是由偏微分方程（PDE）与随机系数依赖于微观尺度描述的异质性。随机系数考虑了物理环境的所有可能实现，并且通过施加某些假设，可以预期，平均而言，几乎所有这些解决方案都表现出相同的有效行为。 这个建议的主要目标是通过丰富的基于随机均匀化的问题源，进一步了解椭圆和抛物偏微分方程。均匀化的研究对理论和应用数学都有贡献：均匀化经常在相关方程的分析中暴露出许多有趣的问题，并且它可以直接用于模拟物理过程。我计划集中我的努力在两个主要类别的椭圆和抛物型偏微分方程：（a）非散度型方程，和（B）反应扩散方程。这些方程分别作为随机控制理论、金融学、几何学和化学动力学、燃烧学和生物学中的主要数学模型。我提出的项目是出于以下两个目标：（1）扩大类的偏微分方程的均匀化发生和（2）获得更具体的信息，均匀化的过程中，如误差估计和性能的有效行为。均匀化的研究结合了数学的几个不同领域的工具，包括分析，偏微分方程，动力系统和概率。我致力于使用合作的方法为拟议的研究计划;从各个子领域汲取灵感和技术。这种灵活的视角促进了对物理现象的统一理解，并增强了非发散形式和反应扩散方程的理论。此外，在这一特定的研究计划的进展可能会影响上述相关领域的数学的发展。除了直接应用到其他数学领域，多尺度问题的研究一直是感兴趣的专家在几个外部领域，包括材料科学，化学工程和生物学。因此，这项工作有助于加强偏微分方程的数学理论和其他科学学科的应用之间的关系。

The theory of stochastic homogenization identifies the average, macroscopic behavior of a phenomenon which is subject to microscopic, random effects. For example, one may be interested in determining the general properties of a porous material with randomly distributed impurities, or predicting the evolution of a population in a heterogeneous medium with random obstacles. Such phenomena are typically modeled by partial differential equations (PDEs) with random coefficients which depend on microscopic lengthscales describing the heterogeneities. The random coefficients take into account all possible realizations of a physical environment, and by imposing certain hypotheses, one may expect that asymptotically on average, almost all such solutions exhibit the same effective behavior. The main goal of this proposal is to further the general understanding of elliptic and parabolic PDEs through the rich source of problems based in stochastic homogenization. The study of homogenization offers contributions to both theoretical and applied mathematics: homogenization often exposes many interesting problems in the analysis of the relevant equations, and it can be directly used to model physical processes. I plan to focus my efforts on two main classes of elliptic and parabolic PDEs: (a) Nondivergence Form Equations, and (b) Reaction-Diffusion Equations. Such equations serve as the primary mathematical models in stochastic control theory, finance, and geometry and chemical kinetics, combustion, and biology respectively. The projects I am proposing are motivated by the following two objectives: (1) To broaden the class of PDEs for which homogenization takes place and (2) To obtain more specific information about the process of homogenization, such as error estimates and properties of the effective behavior. The study of homogenization combines tools from several different areas of mathematics, including analysis, PDEs, dynamical systems, and probability. I am committed to using collaborative approaches for the proposed research program; drawing inspiration and techniques from various subfields. This flexible perspective promotes a unified understanding of the physical phenomena, as well as enhancing the theory for both nondivergence form and reaction-diffusion equations. Furthermore, progress in this specific research program may influence developments in the above related areas of mathematics.Aside from the immediate applications to other subfields of mathematics, the study of multiscale problems has been a source of interest for specialists in several outside areas including materials science, chemical engineering, and biology. Consequently, this work contributes towards strengthening the relationship between the mathematical theory of PDEs and applications to other scientific disciplines.