Variational approach to the regularity of the free boundaries
2020 - 2025
ERC Starting Grant 2019
A free boundary problem is a boundary value problem
that involves partial differential equation on a domain whose boundary is free,
that is, it is not a priori known and depends on the solution of the PDE itself.
These problems naturally arise in many different models in Physics, Engineering and Economy.
A typical example is a block of melting ice; in this case, the free boundary is the surface
of the ice, the PDE is the heat equation and its solution (the state function) is the temperature distribution.
In this project, we study free boundary problems from a purely theoretical point of view.
The focus is on the regularity of the free boundaries arising in the
context of variational minimization problems as,
for instance, the one-phase, the two-phase and the vectorial Bernoulli problems;
the obstacle and the thin-obstacle problems.
The aim is to develop new techniques for the analysis of the fine structure of the free boundaries,
especially around singularities. Many tools and methods developed in this context
can find application to other problems and domains, including shape optimization problems, area-minimizing surfaces, harmonic maps,
free discontinuity problems, parabolic and non-local free boundary problems.
Keywords: free boundary regularity, the one-phase Bernoulli problem, Alt-Caffarelli,
the two-phase Bernoulli problem, Alt-Caffarelli-Friedman, the vectorial Bernoulli problem,
the obstacle problem, the Signorini problem, epiperimetric inequality,
logarithmic epiperimetric inequality, monotonicity formulas.
• Lecture notes on the regularity of the one-phase free boundaries: pdf .
• The vectorial Bernoulli problem: MTV, CSY, KL, KL2, MTV2.
A call for a one-year post-doc position is open now.
Instructions for applicants are available
Joseph Feneuil (post-doc from 1 September 2021 to 9 July 2022).
Starting from 2022 the seminars and the meetings of the workgroup of the project are held in the Deparment of Mathematics (University of Pisa)
with the participation of the members of the workgroup and the visitors invited on the project.
The main topics are regularity theory, free boundary problems, calculus of variations, elliptic PDEs, geometric analysis.
The dates of the meetings and other notifications are diffused on the mailing list of the workgroup.
06/12/2022. Carlo Gasparetto (SISSA).
A short proof of Allard's theorem.
Abstract. Allard's theorem roughly states that a minimal surface,
that is close enough to a plane, coincides with the graph of a smooth function
which enjoys suitable a priori estimates. In this talk we will show
how one can prove this result by exploiting viscosity technique and a
weighted monotonicity formula.
This talk is based on a joint work with Guido De Philippis and Felix Schulze.
11/10/2022 (cancelled). Hui Yu (National University of Singapore).
Rate of blow-up in the thin obstacle problem.
Abstract. The thin obstacle problem is a classical free boundary problem
arising from the study of an elastic membrane resting on a lower-dimensional
obstacle. Concerning the behavior of the solution near a contact point between
the membrane and the obstacle, many important questions remain open.
In this talk, we discuss a unified method that leads to a rate of convergence
to `tangent cones' at contact points with integer frequencies in general
dimensions as well as 7/2-frequency points in 3d.
This talk is based on recent joint works with Ovidiu Savin (Columbia).
25/05/2022. Salvatore Stuvard (Università degli Studi di Milano).
Existence of canonical multi-phase Brakke flows.
18/05/2022. Edoardo Mainini (Università di Genova).
Linearization of finite elasticity.
Abstract. We discuss the linearization of finite elasticity by means of Gamma-convergence for the case of pure traction problems.
For hyperelastic bodies subject to equilibrated force fields, we show that the limit energy can be different from the global energy
of linear elasticity subject to the same force field, unless suitable conditions are fulfilled.
We also discuss linearization under incompressibility constraint.
12/04/2022. Cristiana De Filippis (University of Parma).
Nonuniform ellipticity and nonlinear potentials.
Abstract. Nonuniform Ellipticity and Nonlinear Potential Theory are two classical topics in the analysis of Partial Differential Equations.
In this talk I show how those themes merge to solve the longstanding open problem of deriving Schauder estimates
for minima of functionals (resp. solutions to elliptic equations) featuring polynomial nonuniform ellipticity.
This is joint work with Giuseppe Mingione (University of Parma).
06/04/2022. Dario Mazzoleni (University of Pavia).
L2-Gradient Flows of Spectral Functionals.
09/03/2022. Nicola Soave (Politecnico di Milano).
Free boundary problems in the spatial segregation of competing systems.
Abstract. In this talk we present some results concerning the spatial segregation in systems with strong competition.
In particular, we focus on two different (but strongly related) issues: long-range segregation models and systems characterized by asymmetric diffusion.
The content of the talk is part of ongoing project with H. Tavares, S. Terracini and A. Zilio.
21/02/2022. Giulia Bevilacqua (Politecnico di Torino).
The Kirchhoff-Plateau problem.
Invited speakers and seminars before 2022
04/12/2022 - 08/12/2022. Carlo Gasparetto (SISSA)
30/11/2022 - 02/11/2022. Gianmaria Verzini (Politecnico di Milano)
7/11/2022 - 18/11/2022. Carlo Gasparetto (SISSA)
11/10/2022 - 15/10/2022. Hui Yu (National University of Singapore)
22/9/2022 - 25/9/2022. Max Engelstein (University of Minnesota)
22/5/2022 - 27/5/2022. Salvatore Stuvard (Università di Milano)
16/5/2022 - 19/5/2022. Edoardo Mainini (Università di Genova)
10/4/2022 - 13/4/2022. Cristiana De Filippis (Università di Parma)
3/4/2022 - 8/4/2022. Dario Mazzoleni (Università di Pavia)
8/3/2022 - 10/3/2022. Nicola Soave (Politecnico di Milano)
20/2/2022 - 23/2/2022. Giulia Bevilacqua (Politecnico di Torino)
16/1/2022 - 29/1/2022. Mickaël Nahon (Université de Savoie, Chambery)
5/12/2021 - 9/12/2021. Luca Spolaor (UC San Diego)
25/10/2021 - 28/10/2021. Alessandro Audrito (ETH, Zürich)
10/9/2021 - 24/9/2021. Ekaterina Mukoseeva (University of Helsinki)
4/7/2021 - 10/7/2021. Mickaël Nahon (Université de Savoie, Chambery)
Title: Variational approach to the regularity of the free boundaries
Duration: 60 months
Starting date: 1 June 2020
Primary coordinator: Bozhidar Velichkov
Host institution: Università di Pisa
Funding Agency: European Research Council
Funding Scheme: Starting Grant
Call year: 2019 Panel: PE1
Project number: 853404
Reference: ERC-2019-StG 853404 VAREG
Acknowledge as: "This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853404)"
Acknowledgements. The project proposal was conceived and written at
Laboratoire Jean Kuntzmann (Université Grenoble Alpes),
where the PI was Maître de Conférences from 2014 to 2019.
It is a pleasure to acknowledge LJK, UGA and the projects ANR CoMeDiC and GeoSpec, for supporting the PI's research in this period,
the team of Fostering Science for the support during the preparation of the proposal.