Partial Differential Equations related to the p-Laplacian

DMS-9970687摘要在这一建议中，PI考虑了偏微分方程理论中的问题，这些问题是由非线性弹性、图像插值、控制理论和经典函数理论中的思想相互作用提出的。 图像插值和非线性弹性力学的最新发展可以表述为具有有界导数的函数的延拓问题。通常，一个函数只在定义域的一个子集上是已知的，问题是如何将它插值或延拓到整个定义域上，并保持最大可能的正则性。这种推广是通过求解一个椭圆边值问题得到的，而用标准的椭圆偏微分方程（包括Laplacian）求解这些问题，总是会导致边界处正则性的丧失。 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。这些算子的原型是所谓的“无限拉普拉斯算子”。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对应于非二次能量的非线性方程。如果我们试图最小化能量的最大值，比如说找出 是最大电压或应力，所得到的方程是一个非常非线性的性质，需要一个不同于传统的线性理论的代数处理。PI的提出继续了这些方程的基本理论的发展，在可能的应用中，我们有图像插值，图像处理的基本操作之一，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.