Singular Higher-order Linearized Monge-Ampère Type Equations with Drifts

Indiana University, Bloomington IN

Investigators

Abstract

This project studies several nonlinear partial differential equations that are relevant to applications in science, economics, engineering, meteorology, and physics. They also have deep connections and applications in several areas of mathematics such as analysis, geometry, numerical methods, and the calculus of variations. For example, among the equations investigated in this project, the semigeostrophic equations are used in weather forecasting, while singular affine maximal surface equations and singular Abreu equations are used in the numerical simulations for the monopolist's problem in economics (where the monopolist needs to design the product line together with a price schedule so as to maximize the total profit) and for nonlinear diffusion and crowd-motion models. The project particularly covers various linearized Monge-Ampère equations with drifts where key structural quantities could be possibly extremely large (singular) or extremely small (degenerate), thus rendering them significantly challenging. Despite their important role in applications, these equations are still poorly understood. The project aims at discovering new underlying principles and developing innovative tools to systematically tackle fundamental problems in this area. Their solutions are expected to reveal the interconnectedness of analysis, partial differential equations, the calculus of variations, mathematical economics, and complex geometry, thereby stimulating interactions among these areas. The results of this project will be disseminated through publications of mathematical research papers and lecture notes and via presentations at national and international venues. An important educational component of this project includes the mentoring of graduate students and attracting undergraduate students to mathematical research. This project, in the field of analysis and partial differential equations (PDE), focuses on the solvability, regularity properties, and asymptotic behavior of singular higher-order linearized Monge-Ampère (LMA) type equations with drifts that arise naturally in complex geometry, meteorology, economics, elasticity, physics, and the calculus of variations with a convexity constraint. The project consists of three main themes. The first one investigates the solvability in higher dimensions of singular affine maximal surface equations and singular Abreu equations with drifts. These are fourth-order equations which can be rewritten as systems of a Monge-Ampère equation and a linearized Monge-Ampère equation. The second theme aims at establishing higher-order derivative estimates for singular LMA equations with certain twisted structures (which allow for changing the solution nature under suitable transformations). The third theme studies the solvability of singular Abreu equations with degenerate boundary data. The principal investigator and his collaborators have recently developed new PDE techniques in the LMA type equations with and without drifts, including new perspectives on perturbing away the potential singularities by combining techniques of both divergence form equations via energy functionals and nondivergence form equations. They are expected to be further developed to successfully attack the problems in this project, thereby bringing fresh insights into the study of nonlinear PDE and providing novel approaches to regularity theory. 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.