Partial Differential Equations 2 (PDE2)

This advanced course on partial differential equations will be centered around methods for obtaining a priori estimates for elliptic equations, with applications to existence and uniqueness for such equations and proving regularity (such as differentiability) of their solutions.

We will in the main part of the course be dealing mostly with linear equations with variable coefficients, but always with a view towards applications to nonlinear equations (also with a few examples of this), giving proofs of all statements in high detail.

In the final part of the course, we will branch out with a lighter treatment of some modern advanced broader topics in PDEs.

Main Part (first 6 weeks):

• Quick review of "PDE" material: Laplace equation, harmonic functions, second order elliptic equations in divergence form, Green’s functions, Poincaré-Sobolev Inequality.
• Hopf's maximum principle, Serrin's comparison principle.
• Gradient estimates via Bernstein's method (cut-offs and maximum principles).
• Morrey’s and Campanato’s lemmas.
• De Giorgi-Nash-Moser's iteration argument.
• Moser's Harnack Inequality.
• Regularity of weak solutions of second order elliptic equations in divergence form.
• Hilbert's 19th Problem vs. De Giorgi's & Nash's solutions to it.
• Nonlinear elliptic equations.
• Alexandrov's maximum principle.
• Alexandrov–Bakelman–Pucci estimates.
• Method of moving planes, Gidas-Ni-Nirenberg theory.
• Krylov–Safonov Hölder estimates and Harnack inequality.

Tapas of Topics in PDEs (last 2 weeks):

• Nonlinear Schrödinger equations.
• Quasi-linear heat equations.
• Blow-up behavior at singular times for semilinear evolution equations: heat, wave and Schrödinger equations.
• Examples from active research areas at MATH within PDEs, e.g. problems from quantum theory, mean curvature flow and variational problems in geometry such as minimal surfaces.