Partial Differential Equations related to the p-Laplacian

与 p-拉普拉斯相关的偏微分方程

• 批准号: 9970687
• 负责人: Juan Manfredi
• 金额: $ 5.04万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970687&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; related; Laplacian

DMS-9970687ABSTRACTIn this proposal, the PI considers problems in the theory of partial differential equations suggested by the interplay of ideas in nonlinear elasticity, image interpolation, control theoryand classical function theory. Recent developments in image interpolation andnonlinear elasticity can be formulated as extension problemsfor functions that have bounded derivatives. Typically, a functionis known only on a subset of its domain of definition.The problem is to interpolate or extend it to the whole domainmaintaining the maximum possible regularity. This extension isobtained by solving an elliptic boundary value problem.The solution of these problems by standard elliptic partialdifferential equations, including the Laplacian, always results in a loss of regularity at the boundary. To preserve the boundedness of the derivatives itis necessary to use nonlinear elliptic equations that are highlydegenerate, whose mathematical theory has just begun tobe understood. The prototype of these operators is the socalled "infinite-Laplacian". In many problems in physics and engineering one has to minimizethe average of a quadratic energy. The equations so obtainedare linear and are useful to describe small perturbations inmany cases. To describe large deviations, often one has toconsider nonlinear equations corresponding to non-quadraticenergies. If we try to minimize themaximum of the energy, say to find out where is the maximumvoltage or stress, the resulting equations are of a verynonlinear nature that requires a different mathematicaltreatment than in the traditional linear theory. The PI proposesto continue the development of the basic theory of these equations.Among the possible applications we have image interpolation,one of the basic operation in image processing, and non-linear elasticity.

在这个方案中，PI考虑了非线性弹性、图像插值、控制理论和经典函数理论的相互作用所提出的偏微分方程理论中的问题。图像插值和非线性弹性的最新发展可以表述为导数有界的函数的扩展问题。通常，一个函数只在其定义域的一个子集上是已知的。问题是将其插值或扩展到整个域，以保持最大可能的规律性。该推广是通过求解椭圆边值问题得到的。用标准椭圆型偏微分方程（包括拉普拉斯方程）求解这些问题，总是导致边界处的正则性丧失。为了保持导数的有界性，必须使用高度退化的非线性椭圆方程，其数学理论才刚刚开始被理解。这些算子的原型就是所谓的“无限拉普拉斯算子”。在许多物理和工程问题中，人们必须最小化二次能量的平均值。这样得到的方程是线性的，在许多情况下对描述小扰动是有用的。为了描述大的偏差，通常必须考虑与非二次能量相对应的非线性方程。如果我们试图最小化能量的最大值，比如找出最大电压或应力在哪里，得到的方程就具有非常非线性的性质，需要与传统线性理论不同的数学处理。PI建议继续发展这些方程的基本理论。在可能的应用中，我们有图像插值，图像处理中的基本操作之一，和非线性弹性。