Advanced PDEs

(3 Credits)

Elliptic regularity: Harmonic functions and Harnack Inequality: Liouville's theorem, removable singularity, Harnack convergence theorems. Cacciopolli inequality and some of its consequences. De Giorgi-Nash Theory. Hyperbolic equations: Local existence and regularity of nonlinear problems, conserved quantities, vector fields method. Further topics, such as variational methods, homogenization, dispersive PDEs