Partial Differential Equations III

In the class we will discuss several selected topics in the theory of partial differential equations. In particular, the leading theme of the course is the study of classical minimal surface theory, i.e. smooth surfaces in the 3-dimensional Euclidean space with vanishing mean curvature.
Among the topics that will be covered there are:

1. Minimal surface equation.
2. Second variation formula, Morse Index and Stability.
3. Weierstrass' representation.
4. Berstein's theorem.
5. Simons' inequality and curvature estimates.
6. Conformal maps and Douglas-Rado solution to the Plateau problem.

Date and time info
Monday 15.00 - 17.00; Friday 10.00 - 12.00

Keywords
Classical minimal surfaces, conformal maps, curvature estimates, elliptic regularity

Prerequisites
Analysis I, II, III and basics PDE

Language
English

Remarks and notes
The level of the class is suited for students from the 7th semester on up to PhD students: indeed, although the tools and the techniques of the course will be almost always elementary (basic analysis and PDEs), the results discussed will have several contact points with recent research developments.