Partial Differential Equations related to the p-Laplacian

在这项建议中，PI考虑了由非线性弹性、图像内插、控制理论和经典函数论中的思想相互作用而提出的偏微分方程组理论中的问题。图像内插和非线性弹性的最新发展可以表示为具有有界导数的函数的延拓问题。通常，一个函数只在其定义域的一个子集上是已知的，问题是如何在保持最大可能正则性的情况下对其进行内插或将其扩展到整个定义域。这种推广是通过求解一个椭圆型边值问题得到的，用标准的椭圆型偏微分方程解这些问题，包括拉普拉斯方程，总是导致在边界上失去正则性。为了保持导数的有界性，需要使用高度退化的非线性椭圆型方程，其数学理论才刚刚开始被理解。这些算符的原型是所谓的“无限拉普拉斯”。在物理和工程中的许多问题中，人们必须使二次能量的平均值最小化。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，人们常常不得不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，比方说找出最大电压或应力在哪里，所得到的方程是非常非线性的，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，我们有图像内插，图像处理中的基本运算之一，和非线性弹性。

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.