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

偏微分方程(PDE)是涉及未知多变量函数f的导数的泛函方程。这些方程通过牛顿物理定律与物理系统的行为密切相关，因此与所有科学(重点是物理学)密切相关。物理中常见的偏微分方程组有：热方程、波动方程、拉普拉斯方程和薛定谔方程。与所提出的研究相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(通常由一个应用于向量值函数的矩阵给出)，则该方程是散度形式的。如果刚才提到的矩阵是非负定的，这个方程称为退化椭圆型。当研究散度形式的椭圆型偏微分方程组时，一个有趣的主题立刻变得明显起来。为了研究光滑系数的非线性椭圆型方程，必须了解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型方程，必须理解具有粗糙(可能不连续)系数的线性椭圆型方程。本文研究的是非线性退化椭圆型方程，右端消失的Monge-Ampere方程就是一个常见的例子。近年来，简并的Monge-Ampere方程和简并的椭圆型偏微分方程组之间建立了联系。在这种情况下，存在一个平行的主题，通过理解退化的椭圆型拟线性方程及其线性对应关系，可以理解Monge Ampere方程。拟议的研究试图为这类方程发展一种完整的理论，解决解的存在性和正则性问题。这将通过与新定义的退化Soblev空间和与之相关的演算相关的新技术来实现。这一计划产生的理论将包括由Serrin，Trudinger等人发展的经典理论。扩展我们的科学知识，实际上是我们宇宙的本质。

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.