Local Regularity of the Complex Monge-Ampère Equation

Local Regularity of the Complex Monge-Ampère Equation Yu Wang In this thesis, we study the local regularity of the complex Monge-Ampère equation, (√ −1∂∂̄u )n = fdx where √ −1∂∂̄u stands for the complex Hessian form and dx the Lebesgue measure. The underline idea of our work is to consider this equation as a full-nonlinear equation and apply modern theory and techniques of elliptic PDEs. Our main results include • A simplified viscosity theory on the solvability of the Dirichlet problem of the complex Monge-Ampère equation. • A small perturbation result: if f is slightly better than Dini continuous and the solution u is L∞-close to a quadratic polynomial whose complex Hessian has determinant 1, then u is C at the points x on which f(x) = 1. • A Liouville type theorem: if u solves (√ −1∂∂̄u )n = dx on entire C and u − 1 2 |x| is of sub-quadratic growth at infinity, then u differs from 1 2 |x| by a linear function. • A converging theorem: Assume f ≥ λ > 0, if a sequence of solutions uk converging uniformly to a smooth solution φ, then uk converges smoothly to φ. • An C-regularity theorem: if f is Hölder and the solution u is in W 2,p for p > n(n− 1), then u is C.