Fully Nonlinear Elliptic Equations

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The research activities of this project will continue to deepen and broaden our understanding of two intimately connected mathematical fields: partial differential equations and differential geometry. The project will have an impact in the study of 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, and of maximal surface systems, which have the roots in general relativity. These Hessian equations are also related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape. The project provides research training opportunities for graduate students. The objectives for special Lagrangian equations are to derive Schauder and Calderón-Zygmund estimates for equations with critical and supercritical phases, to answer whether any homogeneous order two solution in dimension five or higher is trivial, to study low regularity of continuous viscosity solutions to the equations with subcritical phases, to investigate the existence and uniqueness of solutions to the Dirichlet problem for the special Lagrangian equation with continuous variable phase, and to resolve periodic Liouville problems with constraints as well as (entire) Liouville problem for the complex version of the special Lagrangian equation. The aim for symmetric sigma-k equations is to investigate Hessian estimates and regularity for sigma-2 equations in dimension four and higher, to obtain Schauder and Calderón-Zygmund estimates for 3-d sigma-2 equations, and to study the Liouville problem for sigma-k equations. The plan for complex and real Monge-Ampère equations is 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 a noncollapsing condition. For the case of maximal surface systems the goal is to study the Bernstein problems for exterior solutions and regularity for solutions under a noncollapsing condition. The project will also take on Hamiltonian stationary equations, where it aims to establish existence of the solutions to the second boundary value problem and rigidity for the Hamiltonian stationary equations.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.