A Priori Estimates for Linear and Nonlinear Partial Differential Equations

DMS-9970367元这个项目集中在线性和非线性偏微分方程的一些估计及其应用。它包括三个主要的研究活动领域。第一部分主要研究抛物型方程加倍性的一些应用和推广。重点讨论了二阶抛物型方程正解在边界附近的行为。 在第二部分中，我们的目标是在较弱的条件下得到完全非线性方程的正则性估计。对于完全非线性方程组，在随机假设下的先验估计已经得到了很好的发展，对前者的理解将给随机对策论、校准几何和非线性弹性理论提供一些应用。第三部分研究了不等周不等式的最佳常数。在具有标准平坦度量的欧氏空间中存在一个尖锐的等周不等式。一个有趣的问题是在负弯曲空间中的对应问题。肯定的答案也会在负弯曲空间中给出尖锐的Sobolev不等式。这个项目的动机是对我们的自然和社会有一些定性和定量的信息。由于偏微分方程模型甚至描述了我们世界中的许多现象。偏微分方程解的估计是我们所需要的。研究这类方程具有重要的理论意义和实际意义，在传热、化学反应、多孔介质、交通流、生物学和经济学等领域有着广泛的应用。

DMS-9970367YuanThis project concentrates on some estimates for linear and nonlinearpartial differential equations and their applications. It consists ofthree main areas of research activity. In part one, the objective is to investigate some applications and generalizations of doubling propertyfor parabolic equations. The emphasis is on the behavior near the boundaryof positive solutions of second order parabolic equations. In part two,the goal is to derive some regularity estimates for fully nonlinearequations with weaker (than concavity) condition. The a priori estimatefor fully nonlinear equations with concavity assumption is well developed.The understanding of former would give some applications tostochastic game theory, calibrated geometry, and nonlinear elasticitytheory. In part three, the aim is to study the best constant in anisoperimetric inequality. There is a sharp isoperimetric inequality in theEuclidean space with the standard flat metric. An interesting question isthe corresponding one in negatively curved space. An affirmative answerwould also give the sharp Sobolev inequality in the negatively curvedspace.The motivation of this project is to have some qualitative and quantativeinformation about our nature and society. Since partial differentialequations model and even describe many phenomena in our world. Estimatesfor solutions of partial differential equations are what we need. Studyingthe equations posed in this project is important for both practical andtheoretical purposes, and has broad applications in many fields, such asheat transfer problems, chemical reaction, porous media, traffic flows,biology, and economics.