Free boundary problems and related topics

Workshop 4–8 July 2022

Free boundary problems arise in many applied sciences such us Engineering, Finance and Fluid Dynamics: they are usually described by some partial differential equations that exhibit, in addition, some unknown interfaces on which extra boundary conditions are posed. A classical example is the Stefan problem, which describes the melting of an ice block into water. In such class of problems, the main goal is to describe the geometry and regularity of both the solutions and their free boundaries: the mathematical theory involves methods and techniques from Calculus of Variations, Geometric Measure Theory and PDEs, and presents interesting connections with the theory of Minimal Surfaces and Geometric Analysis.

The workshop is structured in three courses of 6 hours each and is aimed to introduce part of the classical theory as well as some of the most recent developments. The participation of PhD students and postdocs is strongly encouraged.

Location

The workshop will take place in the Herman-Weyl-Zimmer (HG G 43) in the external pageETH Main Building at Rämistrasse 101, 8092 Zürich

Speakers and courses

Bozhidar Velichkov (Università di Pisa)

Title: Free boundary regularity for the one-phase Bernoulli problem

Abstract: In these lectures we will go through the basic regularity theory of free boundary problems of Bernoulli type.We will use the Alt-Caffarelli variational formulation and we will focus on the regularity theory in low dimensions. We will also introduce tools as monotonicity formulas, density estimates, viscosity solutions, and epiperimetric inequalities.

Hui Yu (National University of Singapore)

Title: Introduction to the thin obstacle problem

Abstract: The thin obstacle problem describes the shape of an elastic membrane over a lower dimensional obstacle. This is a problem with a lot of history, but many fundamental questions remaining open about the contact set. In this course, starting from the basics, we will review some classical results about the regularity of the solution and the free boundary. In the later half of the course, we will see some new developments about rate of blow-up at contact points.

Zihui Zhao (University of Chicago)

Title: Quantitative unique continuation for elliptic PDEs

Abstract: Unique continuation property is a fundamental property of harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes to infinite order at a point, it must be zero everywhere. In the same spirit, we can use the local growth rate of harmonic functions to deduce important global information, such as estimating the size of their nodal sets or singular sets. I will start the mini-course by introducing examples, useful tools and prior results of quantitative unique continuation. Then I will talk about a few recent results and explain the key ideas of their proofs.