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

点击翻译按钮获取中文摘要

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.