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

偏微分方程（PDEs）是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（重点是物理学）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Schrödinger方程。与所提出的研究相关的一类方程是散度形式的退化椭圆方程。如果一个方程的最高阶微分项是由一个向量场的散度给出的，那么这个方程就是散度形式的（通常是由一个矩阵作用于一个向量值函数给出的）。如果上述矩阵是非负定的，则称为简并椭圆型方程。当研究发散形式的椭圆偏微分方程时，一个有趣的主题立即显现出来。为了研究具有光滑系数的非线性椭圆方程，必须理解具有连续系数的拟线性椭圆方程。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（可能是不连续）系数的线性椭圆方程。本文主要研究非线性退化椭圆方程；右手边消失的蒙日-安培方程就是一个熟悉的例子。近年来，人们建立了简并蒙日-安培方程与简并椭圆偏微分方程之间的联系。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椭圆拟线性方程及其线性对应物来理解蒙日安培方程。拟议的研究旨在为这些方程建立一个完整的理论，以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空间和与之相关的微积分相关的新技术来实现。从这个项目中产生的理论将包括塞林、特鲁丁格等人发展的经典理论，扩展我们的科学知识，实际上是宇宙的本质。

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.