This is the homepage of Min-max theory and minimal surfaces. I will teach this course during July 6-July 10. The course notes, some extra material, the meeting link and exercises for the course are posted below.

Logistics of the course

We will have an hour of lecture per day. We will discuss the exercises through the Slack channel of the mini-course.

• Lecture time: 11:00 am - 12:00 pm (US central time), Monday July 6- Friday July 10.

• Zoom information:

Topic: Min-max theory

Join Zoom Meeting

https://utexas.zoom.us/j/7485529452?pwd=TWNkVHAzUmZyN252Y0EyV0Zta2hlZz09

Meeting ID: 748 552 9452

Password: Morse

Abstract

This course is designed for people with some background in variational methods in analysis and familiar, in some sense, with the basic notions in geometric analysis (perimeter, BV functions, Hausdorff measures, coarea formulas and related concepts). Nonetheless, in the first part, we will start reviewing basic concepts of Sobolev spaces, BV spaces, Calculus in Banach spaces and the direct method. The second part is devoted to the study of the main tool in min-max theory: the deformation lemma. Time provided, we will discuss the link between min-max theory, Schauder degree and Morse theory. In the third session, we will start exploring some applications in semilinear PDE including the Allen-Cahn equation, which will serve as a link between minimal surfaces and this theory. Fourth and fifth sessions will be focused in the applications to the Almgren-Pitts theory recently developed by Fernando C. Marques and André Neves (and their students) as a limiting case of the classical results in Min-max theory.

I am (currently) typing some notes for this mini-course. The version will be updated repeatedly throughout this week and can be downloaded from here.

Outline

Day 1 (July 6)

Calculus in Banach spaces, IVP in Banach spaces, quick review of Sobolev spaces and related topics.

This first section summarizes the content of these other notes I wrote long time ago, which in turn heavily borrow ideas from Arbogast and Bona nodes in Applied Math.

You can find the recorded meeting in here.

Day 2 (July 7)

Deformation lemma in Banach spaces, some generalizations and applications: Mountain-Pass theorem, Saddle-Point theorem, etc.

There are many good references for this topic. I am following closely Paul Rabinowitz notes, the book of Chang in Infinite dimensional Morse theory and this book of Ghoussub.

You can find the recorded meeting in here.

We will review briefly the properties of the spectrum of the negative Laplacian in bounded domains and how to verify the Palais-Smale condition in this case. Then, we will discuss some applications of the min-max theorems to obtains one signed solutions are least energy nodal solutions in the Dirichlet case. Whilst, in the Neumann case we will see how to combine Leray-Schauder degree with the Mountain-Pass theorem to obtain multiple solutions. These notes of Manuel del Pino and translated by Fernando Morales are a good reference for Degree theory in finite dimension.

This part will be based on the following papers:

• Castro, A., Cossio, J., & Neuberger, J. M. (1997). A sign-changing solution for a superlinear Dirichlet problem. The Rocky Mountain journal of mathematics, 1041-1053.

• Barile, S., & Figueiredo, G. M. (2014). Existence of a least energy nodal solution for a class of p&q-quasilinear elliptic equations. Advanced Nonlinear Studies, 14(2), 511-530.

• Agudelo, O., Correa, S., Restrepo, D., & Vélez, C. (2018). Multiplicity results and qualitative properties for Neumann semilinear elliptic problems. Journal of Mathematical Analysis and Applications, 468(1), 141-160.

• Aftalion, A., & Pacella, F. (2004). Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. Comptes Rendus Mathematique, 339(5), 339-344.

You can find the recorded meeting in here.

Day 4 (July 9)

Continuation of lecture 3.

Day 5 (July 10)

Application to the study of the existence of min-max solutions for phase transitions. The main references are this paper of Marco Guaraco and these notes from a summer school at Princenton written by him.