Organizers: Ricardo Nochetto, Wujun Zhang
When: Wednesdays @ 5pm-6pm, starting the first week of February
Where: room TBA
Subtitle: PDE theory and numerical analysis

Fully nonlinear second order elliptic PDEs arise naturally from differential geometry, stochastic control theory, optimal transport and other fields in science and engineering. In this RIT, we will discuss the concept of viscosity solutions and regularity theory of these PDEs. Some possible topics include:

• fully nonlinear elliptic equations and viscosity solutions
• Alexandroff-Bakelman-Pucci estimates
• Harnack inequality and Hölder regularity
• Uniqueness of solutions
• W^2,p estimates
• C^{1, α} estimates
• C^{2, α} estimates

In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. We would like to discuss some criteria for designing convergent numerical methods and some tools developed recently which are useful to obtain rates of convergence. Some possible topics include:

• criterion for convergence of numerical methods
• discrete version of the Alexandroff-Bakelman-Pucci estimate
• finite element method for linear elliptic equations in non-divergence form
• numerical method for Monge-Ampere equations