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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573821
• 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.

部分微分方程（PDE）是涉及未知多变量函数f的衍生物f的功能方程。这些方程与所有科学密切相关（重点是物理学），因为它们通过牛顿的物理定律与物理系统的行为联系在一起。物理学上的一些熟悉的PDE是：热方程，波动方程，拉普拉斯方程和Schrödinger方程。与拟议的研究相关的方程式类别是变性形式的退化椭圆方程。如果方程式由矢量场的差异给出（通常由应用于向量值函数的矩阵给出），则方程式为差异形式。如果刚刚提到的矩阵是未定义的，则该方程称为退化椭圆形。当研究椭圆形PDE形成椭圆时，一个有趣的主题立即显而易见。为了研究具有平滑系数的非线性椭圆方程，必须了解具有连续系数的准椭圆方程。为了理解具有连续系数的准椭圆方程，必须了解具有粗糙（可能是不连续的）系数的线性椭圆方程。提出的研究研究非线性退化椭圆方程； Monge-Ampere方程式具有消失的右侧，这是一个熟悉的例子。近年来，堕落的monge-ampere方程与退化椭圆形PDE之间建立了联系。在这种情况下存在一个平行的主题，并且可以通过理解退化的椭圆形准方程及其线性对应物来实现对Monge Ampere方程的理解。拟议的研究旨在为解决存在问题和解决方案的规律问题开发完整的理论。这将通过与新定义的退化Sobolev空间以及与之相关的计算有关的新技术来实现。该程序产生的理论将包括Serrin，Trudinger等开发的经典理论。 Al。，扩展我们对科学的知识和实际上的宇宙本质。