Regularity of Weak Solutions to Degenerate Nonlinear/Quasilinear Equations with Rough Coefficients

具有粗糙系数的退化非线性/拟线性方程弱解的正则性

• 批准号: 418975-2012
• 负责人: Rodney, Scott
• 金额: $ 1.09万
• 依托单位: Cape Breton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635219
• 关键词: Regularity; Weak; Solutions; Degenerate; Nonlinear

Partial differential equations (PDEs) are functional equations that involve the derivatives of an unknown multivariable function f. These equations are closely connected to all of the sciences (with an emphasis on Physics) due to their connection with the behavior of physical systems through Newton's laws of Physics. Some familiar PDEs in physics are: the heat equations, the wave equation, Laplace's equation and Schrödinger's equation. The class of equations relevant to the proposed research is that of degenerate elliptic equations in divergence form. An equation is in divergence form if its highest order differential terms are given by the divergence of a vectorfield (most often given by a matrix applied to a vector valued function). The equation is called degenerate elliptic if the matrix just mentioned is non-negative definite. When studying elliptic PDEs in divergence form an interesting theme becomes immediately apparent. In order to study non-linear elliptic equations with smooth coefficients one must understand quasilinear elliptic equations with continuous coefficients. In order to understand quasilinear elliptic equations with continuous coefficients one must understand linear elliptic equations with rough (possibly discontinuous) coefficients. The proposed research studies nonlinear degenerate elliptic equations; the Monge-Ampere equation with vanishing right hand side serves as a familiar example. In recent years, a connection has been established between the degenerate Monge-Ampere equation and degenerate elliptic PDEs. A parallel theme exists in this circumstance and an understanding of the Monge Ampere equation may be achieved through the understanding of degenerate elliptic quasilinear equations and their linear counterparts. The proposed research seeks to develop a complete theory for such equations that addresses questions of existence and regularity of solutions. This will be achieved through new techniques related to the newly defined degenerate Sobolev spaces and the calculus connected to them. The theory arising from this program will include the classical theory developed by Serrin, Trudinger, et. al., expand upon our knowledge of the sciences and in effect, the nature of our universe.

偏微分方程（PDE）是一种涉及未知多变量函数f的导数的函数方程。这些方程与所有科学（特别是物理学）密切相关，因为它们通过牛顿物理定律与物理系统的行为联系在一起。物理学中常见的偏微分方程有：热方程、波动方程、拉普拉斯方程和薛定谔方程。与所提出的研究有关的一类方程是散度形式的退化椭圆型方程。如果一个方程的最高阶微分项由向量场的散度给出（通常由应用于向量值函数的矩阵给出），则该方程是散度形式的。如果矩阵是非负定的，则称该方程为退化椭圆方程。当研究椭圆偏微分方程的发散形式，一个有趣的主题变得立即明显。为了研究具有光滑系数的非线性椭圆型方程，必须了解具有连续系数的拟线性椭圆型方程。为了理解具有连续系数的拟线性椭圆方程，必须理解具有粗糙（可能不连续）系数的线性椭圆方程。本文研究了非线性退化椭圆型方程，以右手边为零的Monge-Ampere方程为例。近年来，退化Monge-Ampere方程和退化椭圆型偏微分方程之间的联系被建立起来。一个平行的主题存在于这种情况下，蒙日安培方程的理解可以通过退化椭圆拟线性方程及其线性对应的理解。拟议的研究旨在为此类方程开发一个完整的理论，解决解的存在性和规律性问题。这将通过与新定义的退化Sobolev空间和连接到它们的微积分相关的新技术来实现。从这个计划产生的理论将包括Serrin，Trudinger等开发的经典理论。例如，扩展我们的科学知识，实际上，我们宇宙的本质。