MATH 7120 - Nonlinear Elliptic Equations

Daniel Rubin, spring 2017.

I will discuss the general theory of elliptic equations in non-divergence form, with a view towards second-order fully nonlinear equations. Aspects of the theory include the ABP estimate, viscosity solutions, the Harnack inequality, Schauder estimates, and the Evans-Krylov $C^{2,\alpha}$ estimate for concave equations. Then I will discuss the specific case of the real Monge-Ampere equation $\mathrm{det}(D^2 u) = f$.