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

Harnack不等式、退化演化方程和应用数学的主题建议研究摘要Emmanuele DiBenedetto该项目将继续研究退化椭圆和抛物方程的某些重要类别以及均匀化理论。一个主要的课题将是Harnack不等式的研究。这些不等式已被视为某些类型的线性和非线性偏微分方程的解的性质。这个项目将着眼于证明Harnack型不等式对新函数类的有效性。本研究的一部分将集中在亚临界和超临界状态下抛物型p-Laplacian方程的解的分析。均匀化理论中的问题将集中在模拟分子在脊椎动物视网膜杆外节（ROS）中的扩散。将计算并确定此类扩散过程的均匀化限值，设置在ROS的圆柱形、厚层和切割结构中。数学问题在于识别拓扑和相对紧性，以允许极限过程。 这些结构的技术新奇在于，当层的厚度变为零时，畴变得断开。Harnack不等式在椭圆和抛物型微分方程理论中起着核心作用，并在分析，几何和应用中广泛使用。确定它们的有效性范围可能会影响数学分析中的许多主题。均匀化问题是由视觉过程中的视觉传递引起的。这一分析的结果可能有助于阐明视觉的生物学过程。

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.