Regularity Estimates for the Linearized Monge-Ampere and Degenerate Monge-Ampere Equations and Applications in Nonlinear Partial Differential Equations

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This research project studies fine quantitative behaviors of solutions to several classes of nonlinear partial differential equations (PDEs) that have connections and applications in several areas of mathematics such as analysis, PDEs, the calculus of variations, convex geometry, shape optimization, and fluid mechanics. They also appear in many areas of sciences and engineering such as economics, urban planning, meteorology, and geometric optics. For example, the Monge-Ampere type equations investigated in this project arise naturally in the optimal transportation problems (which consist of finding the least expensive way to transport a distribution of mass from one location to another) in economics and in traffic network planning in cities, in the design of reflector antennae in geometric optics and in the weather forecast models used in meteorology. The PDEs investigated in this project have two distinguished features: their key structural quantities could be possibly extremely small (degenerate) or extremely large (singular) and their settings frequently involve irregular geometries. Classical methods are usually inadequate in handling these equations and thus, their analysis calls for new methods, fresh perspectives and advancing knowledge in many fields of mathematics. The main goal of the project aims at providing deep insights into these problems, discovering novel methodologies to tackle them as well as revealing unexpected connections with other areas of mathematics. The results of this project will be widely disseminated via publications of research papers and lecture notes, via presentations at national and international venues, and via training of graduate students.This project, in the field of analysis and partial differential equations (PDEs), focuses on regularity properties of solutions to the linearized Monge-Ampere (LMA) and degenerate Monge-Ampere equations and their applications in nonlinear PDEs arising from convex geometry, optimal transportation, and meteorology. The purpose of this project is to obtain fine and higher order regularity properties of some important classes of LMA and degenerate Monge-Ampere equations and apply them to several interesting problems in analysis, geometry, and PDEs. More specifically, the objectives of the project are to: (i) investigate higher order derivatives estimates for LMA equations with lower order terms having low regularity and their applications in the semigeostrophic equations as well as polar factorizations; (ii) study the sharp Sobolev estimates for the Monge-Ampere equation and its related maximal functions; (iii) settle a shape optimization problem concerning the minimum of the Monge-Ampere eigenvalue on convex domains subject to a volume constraint; and (iv) establish global regularity for degenerate Monge-Ampere equations on nonsmooth domains and the second boundary value problem for degenerate Monge-Ampere equations. The principal investigator and his collaborators have recently developed new methods and techniques including the Green function estimates, localization technique, iteration argument, sliding paraboloids method and geometry of the Monge-Ampere equation to solve some open problems related to these proposed problems. These techniques are expected to be further developed and strengthened to successfully attack the problems proposed in this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.