Topics in Harnack Inequalities, Degenerate Evolution Equations, and Applied Mathematics

哈纳克不等式、退化演化方程与应用数学主题研究计划摘要本项目将继续研究若干重要的退化椭圆型方程和抛物型方程以及齐次化理论。一个主要的主题将是哈纳克不等式的研究。这些不等式被认为是某些类型的线性和非线性偏微分方程解的性质。这个项目将着眼于证明新的函数类的harnack型不等式的有效性。本研究的一部分将集中于分析抛物型p-拉普拉斯方程在亚临界和超临界状态下的解。均质化理论的主要问题是如何模拟脊椎动物视网膜杆外段（ROS）中分子的扩散。均匀限制将计算和识别这种扩散过程，设置在圆柱形，厚层和切割结构的活性氧。数学问题在于确定拓扑结构和相对紧凑性，以允许极限过程。这些结构在技术上的新颖之处在于，当层的厚度趋于零时，域就会断开。哈纳克不等式在椭圆型和抛物型微分方程的理论中起着中心作用，在分析、几何和应用中得到了广泛的应用。确定它们的效度范围可能会影响数学分析中的许多主题。同质化问题是由视觉过程中的视觉转导引起的。这一分析结果可能对视觉的生物学过程有所启发。

Topics in Harnack Inequalities, Degenerate Evolution Equations and Applied MathematicsAbstract of Proposed ResearchEmmanuele DiBenedettoThis project is to continue research on certain important classes of degenerate elliptic and parabolic equations and on homogenization theory. A major topic will be the study of Harnack inequalities. These inequalities have been regarded as properties of the solutions of certain types of linear and non-linear partial differential equations. This project will look at proving the validity of Harnack-type inequalities for new classes of functions. Part of this study will concentrate on the analysis of solutions of parabolic p-Laplacian equations in both the sub-critical and super-critical regimes. The problems in homogenization theory will center on modeling the diffusion of molecules in the rod outer segment (ROS) in the retina of vertebrates. Homogenized limits will be computed and identified for such diffusion processes, set in the cylindrical, thickly layered and incised structures of the ROS. The mathematical problem consists in identifying a topology and relative compactness to permit the limit process. The technical novelty of these structures is that the domain becomes disconnected as the thickness of the layers goes to zero. Harnack inequalities play a central role in the theory of elliptic and parabolic differential equations and are used extensively in analysis, geometry and applications. Determining their range of validity may impact many topics in mathematical analysis. The homogenization problem is motivated by visual transduction in the process of vision. The results of this analysis may cast light on the biological process of vision.