Fully Nonlinear Elliptic Equations

This project will deepen and broaden our understanding of two closely connected mathematical fields: partial differential equations and differential geometry. The project studies special Lagrangian equations, complex Monge–Ampère equations, and Hamiltonian stationary equations, which provide the mathematical foundation for mirror symmetry in the string theory of modern physics. These equations are also important in nonlinear elasticity and optimal transport. The project provides training opportunities for graduate students and postdoctoral researchers. For special Lagrangian equations, the objectives are to derive (optimal) Schauder and Calderón–Zygmund estimates, regularity and rigidity in dimension five and higher, existence, uniqueness and low regularity with variable continuous general phase, periodic Liouville theorems and existence with constraints, as well as a complex version of rigidity for these equations. The aim of the part of the project devoted to symmetric sigma-k equations is to investigate existence of singular solutions, integrability of the Hessian and partial regularity for the sigma-2 equations in dimension five and higher, to obtain Schauder and Calderón–Zygmund estimates for three- and four-dimensional sigma-2 equations, to study the Liouville problem for sigma-k equations, and to investigate the "double divergence" integral solutions to the sigma-2 equation. The project also aims to demonstrate the triviality of any global solution to complex Monge–Ampère equations, including self-shrinking equations for the Kähler-Ricci flow with certain necessary restrictions and to derive regularity of solutions to the real Monge–Ampère equations under an extrinsic noncollapsing condition. For the case of Hamiltonian stationary equations, the project's goals are to establish rigidity and existence of solutions to the second boundary value problem. 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.