Homogenization of Elliptic and Parabolic Partial Differential Equations

随机均质化理论确定了受微观、随机效应影响的现象的平均、宏观行为。例如，人们可能感兴趣的是确定具有随机分布的杂质的多孔材料的一般性质，或者预测具有随机障碍的非均匀介质中的种群的演化。这种现象通常由具有随机系数的偏微分方程组(PDE)来模拟，这些随机系数取决于描述非均质性的微观长度尺度。随机系数考虑了物理环境的所有可能实现，通过施加某些假设，人们可以预期，平均而言，几乎所有这样的解都表现出相同的有效行为。*本建议的主要目的是通过基于随机齐化的丰富的问题源来加深对椭圆型和抛物型偏微分方程组的一般理解。齐次化的研究为理论和应用数学提供了贡献：在分析相关方程时，齐次化经常暴露出许多有趣的问题，它可以直接用于对物理过程进行建模。*我计划集中精力研究两类主要的椭圆型和抛物型偏微分方程组：(A)无散度型方程和(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、燃烧和生物学的主要数学模型。我提议的项目是出于以下两个目标：(1)扩大发生同质化的偏微分方程的类别；(2)获得关于同质化过程的更具体的信息，例如误差估计和有效行为的性质。*齐次化的研究结合了几个不同数学领域的工具，包括分析、偏微分方程、动力系统和概率。我致力于在拟议的研究计划中使用协作方法；从各个子领域汲取灵感和技术。这种灵活的观点促进了对物理现象的统一理解，并加强了对无散度形式和反应扩散方程的理论。此外，这一特定研究计划的进展可能会影响上述相关数学领域的发展。除了直接应用于数学的其他子领域外，多尺度问题的研究一直是包括材料科学、化学工程和生物在内的几个外部领域的专家的兴趣来源。因此，这项工作有助于加强偏微分方程的数学理论与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.