This is the homepage of Free boundary problems. I will teach this course during July 26-July 30. The course material, the meeting link and some exercises for the course are (or will be) posted below.

Logistics of the course

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

• Lecture time: 11:00 am - 12:00 pm (US central time), Monday July 26-July 30.

• Zoom information:

Topic: Free boundary problems

Join Zoom Meeting

https://utexas.zoom.us/j/96720275723?pwd=Z1N4ZFFLN2kwMmpkVGhlVndmWTJrUT09

Meeting ID: 967 2027 5723

Password: obstacle

Abstract

The aim of this course is two introduce two models in which free boundaries naturally appear: the obstacle problem and the one phase Bernoulli problem. Since the understanding of other more complicated problems in which free boundaries arise are analogous to these two (e.g., Bernoulli 2 phase problems, two obstacle problems, think obstacle problems, their parabolic versions, etc.), we will focus in a systematic study of these two in the most simple case. In the first part, we will start motivating both problems, discussing existence and relevant properties of solutions. The second session will be devoted to understanding the scaling properties of these two problems and their optimal regularity. The last three sessions will bring us to the core of the matter: free boundaries. We will study the regularity of free boundaries through a parallel between them and minimal surfaces, i.e., we will show how techniques and tools such as monotonicity formulas, De Giorgi iterations and blow ups are at the core of the (current) research agenda that seeks to establish their sharp properties.

Core references for the course

Obstacle problem

• Notes by Xavier Ros-Oton. These notes condense pretty much what we are going to see about the obstacle problem and is the main reference for the minicourse regarding this topic.

• Notes by Alessio Figalli. These notes are complementary to Xavi's notes, although more synthetic. They also provide and insightful discussion about the stratification of the singular part of the free boundary.

• Regularity of free boundaries in obstacle-type problems by Uraltzeva, Shahgolian and Petrosyan. This book provides a thorough and rigorous exposition of the topic starting from the basics. This will be our main source for the theory regarding monotonicity formulas.

One phase Bernoulli problem

• Notes by Bozhidar Velichkov. This, combined with Caffarelli-Salsas' book, will be our main source for this topic during the minicourse.

• A geometric approach to free boundary problems by Caffarelli and Salsa. This book contains all the cornerstones in the theory and it is somewhat technical, but readable. Nonetheless, several results of its appendix will be presented in the minicourse, e.g., the boundary Harnack inequality.

Lecture 1

Introduction to free boundary problems and brief presentation of the two problems that we will address throughout the course. Presentation of the general strategy borrowed from the context of minimal surfaces to study the regularity of free boundaries. Notes in here.

Lecture 2

Obstacle problem, brief introduction, optimal regularity and non-degeneracy of solutions. We also discuss the existence of mean value sets for divergence type operators with bounded measurable coefficients (Proved by Blank and Hao). Notes in here.

Lecture 3

One phase Bernoulli problem, brief introduction, optimal regularity (we did not discuss non-degeneracy of solutions). See, for instance, Caffarelli-Salsa Chapter 1. We also discussed the approximation to the regularity and the existence through singular activation (approximating the minimizer by solutions of semilinear PDE). We did not show the convergence of the minimizers of the auxiliary functionals, but it can be shown easily with the Lipschitz bounds derived from the optimal regularity (exercise). Notes in here. (The recording of this session is available and can be sent upon request).

Lecture 4

First part of the study of the regularity of free boundaries. Discussion about the blow-ups (classification of the blow-ups in the case of the Bernoulli one phase problem) and introduction of the blow-ups in the case of the obstacle problem. We also discussed their corresponding monotonicity formulas and the proof of the regularity of the free boundary of the Bernoulli one phase problem up to dimension 4. . Notes in here. (The recording of this session is available and can be sent upon request).

Errata: In the video and in the lecture notes I claimed that the degree of homogeneity of a solution approaches to two as the cone fills the space. This is false. Actually, the degree of homogeneity goes to plus infinity as the angle of the cone goes to zero and goes to 1/2 as the cone becomes "thinner". There is a proof of this fact for the case n=2 on these notes.