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

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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.