Geometric problems in PDEs

Description

Abstract: The interplay between partial differential equations (PDEs) and geometry is a fruitful area of research. It includes from problems arising in geometry and addressed with PDEs and analysis tools to questions concerning solutions to certain PDEs with a geometric description. 

This seminar aims to introduce different questions arising in this area to illustrate its richness at the same time we review some theory of both PDEs and geometry. 

Some of the problems we may deal with include:

• Levels sets of solutions to PDEs.

• Minimal surfaces and the Plateau problem.

• Isoperimetric inequalities for eigenvalues of the Laplacian.

• Geometric inverse problems.

• The Yamabe problem.

Prerequisites: Previous knowledge on differential geometry and PDEs would be desirable.

Bibliography: (partial, more details in the list of topics)

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer-Verlag, Berlin (1998).

X. Cabré et al. Geometry of PDEs and Related Problems, Lecture Notes in Mathematics, Fond. CIME/CIME Found. Subser. 2220, Springer (2018).

Notes

Next you can find some handwritten partial notes I have prepared for the seminar:

📄 General introduction.

📄 Isoperimetric inequalities for eigenvalues of the Laplacian (introduction, Faber-Krahn inequality, further tools and remarks) [T1].

📄 Submanifolds that are level sets of solutions to PDEs (Introduction, general strategy and basic tools) [T6].