A Priori Estimates for Linear and Nonlinear Partial Differential Equations

DMS-9970367元本项目主要研究线性和非线性偏微分方程解的估计及其应用。它包括三个主要领域的研究活动。在第一部分中，我们的目的是研究抛物型方程加倍性质的一些应用和推广。重点研究了二阶抛物型方程正解在边界附近的性态。在第二部分中，我们的目的是在弱(比凹)的条件下得到完全非线性方程的正则性估计。具有凹性假设的完全非线性方程的先验估计得到了很好的发展。对先验估计的理解将在随机博弈论、校准几何和非线性弹性理论中有一定的应用。在第三部分中，目的是研究各向异性不等式的最佳常数。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。一个有趣的问题是负弯曲空间中相应的问题。一个肯定的答案也会给出负曲线空间中尖锐的索博列夫不等式。这个项目的动机是获得关于我们的自然和社会的一些定性和定量的信息。因为偏微分方程组模拟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。研究本课题提出的方程具有重要的理论和实际意义，在传热学、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.