Partial Differential Equations With and Without Convexity Constraints

This project studies fine quantitative properties of selected problems in nonlinear partial differential equations (PDE) and the calculus of variations with and without a structural condition called convexity. These problems have connections and applications in several areas of mathematics such as analysis, PDEs, the calculus of variations, and numerical methods.

Moreover, they are equally relevant to important applications in other areas of science and engineering. For example, the PDE and calculus of variations problems with a convexity constraint investigated in this project arise in different scientific disciplines such as Newton’s problem of minimal resistance in physics, the monopolist’s problem (where the monopolist needs to design the product line together with a price schedule so as to maximize the total profit) in economics, or wrinkling patterns of elastic shells in elasticity.

Despite its ubiquity, the calculus of variations with a convexity constraint is still poorly understood. This project develops new mathematical tools, especially PDE techniques, to better understand basic problems in this area. An important theme of this project is to investigate fundamental questions in one field of PDE and the calculus of variations by importing ideas and developing methodologies from other fields, such as using complex geometric insights to tackle PDE questions arising in economics and physics, or using convexity techniques to understand fine properties of fully nonlinear PDEs without any convexity.

This project provides training opportunities for graduate students, and its results will be disseminated to diverse audiences via publications of research papers and lecture notes and via presentations at national and international venues.

This project focuses on the solvability, regularity estimates, and asymptotic analysis, of several classes of fully nonlinear elliptic PDE and problems in the calculus of variations, with and without convexity constraints and apply them to several interesting problems in analysis and PDE and those arising in economics and elasticity. The project consists of four main parts.

The first one investigates to what extent one can approximate the minimizers (and their Euler-Lagrange equations) of convex functionals with a convexity constraint by solutions of singular Abreu equations which arise in complex geometry. One of such functionals is the Rochet-Chone model for the monopolist's problem in economics. The second part aims to establish the global solvability of highly singular Abreu equations.

These fourth order equations can be rewritten as systems of a Monge-Ampere equation and a linearized Monge-Ampere equation. The third part studies the convergence of an inverse iterative scheme for the k-Hessian eigenvalue problem. The last part investigates the sharp decay, with respect to the ellipticity ratio, for the small integrability exponent in the second derivative estimates for fully nonlinear elliptic equations without convexity, and linearized Monge-Ampere equations.

The principal investigator (PI) aims to systematically develop the Monge-Ampere type equation techniques to study fine properties of fully nonlinear PDE without any convexity. Moreover, recent PDE methods introduced by the PI and his collaborators (such as nonlinear integration by parts for k-Hessian equations and partial Legendre transforms for fourth order equations of Monge-Ampere type) will be further explored to successfully attack the problems to be investigated as part of 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.