Regularity of degenerate elliptic equations and systems, and applications to evolutionary equations and monge-ampere equations

该程序集中在偏微分方程（PDE）的不同领域，其共同点是它们涉及系数“粗糙”或方程不满足某些理想的结构假设（称为椭圆度）的问题，这使得它们的处理更具挑战性。这类偏微分方程在纯科学和应用科学中具有极其重要的意义，因为它们极大地促进了对传热、市场定价、电动力学和天体力学等现象的理解，其中一个目标是为我们开发的用于处理某些非线性方程的强大技术--偏勒让德变换（PLT）找到新的应用。特别地，我们可以处理具有不同退化度的蒙格-安培方程。这些方程与几何中曲率的概念有关，它们模拟了各种各样的应用，包括最佳运输和气象学领域的一些应用。最近，（非退化）椭圆算子解决了突出的“加藤平方根问题”。这一结果可以应用于双曲方程，双曲方程描述了波的运动，因此可以模拟声音、光和量子现象。我们将研究这个问题的退化椭圆算子的推广。本研究的意义源于许多真实的生活现象并不一定满足经典的椭圆性假设，我们还提出研究具有粗糙系数的线性椭圆型方程边值问题的可解性。对这些更基本类型的方程的理解对于非线性方程理论的发展至关重要。

This program is concentrated in different areas of Partial Differential Equations (PDEs), with the common thread that they concern problems where the coefficients are "rough" or where the equations do not satisfy a certain desirable structural assumption (called ellipticity), what makes their treatment more challenging. Such PDEs are of paramount importance in pure and applied sciences, since they greatly contribute to the understanding of phenomena such as heat transfer, market pricing, electrodynamics, and celestial mechanics, among many others.One objective is to find new applications for a powerful technique, the Partial Legendre Transform (PLT), that we developed to treat certain nonlinear equations. In particular, we can treat Monge-Ampere equations with various degrees of degeneracy. These equations are related to the concept of curvature in geometry and they model a wide variety of applications, including some in the areas of optimal transport and meteorology.The prominent "Kato square root problem" was solved recently for (nondegenerate) elliptic operators. This result has applications to hyperbolic equations, which describe the movement of waves and therefore model sound, light, and quantum phenomena. We will study a generalization of this problem for degenerate elliptic operators. The significance of this research derives from the fact that many real life phenomena do not necessarily satisfy the classical assumptions of ellipticity.We also propose to study the solvability of boundary problems for linear elliptic equations with rough coefficients. The understanding of these more basic type of equations is crucial for the evolution of the theory of nonlinear equations.