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.

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 2D case, applications; singularities.

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.

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).

Abstract (pdf).

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.

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).

Abstract (pdf).

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.

Abstract (pdf).

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)

- Summary: Cover letter.

- Summary: Cover letter.

- Oberwolfach report 2020: Regularity of the two-phase free boundaries.

- Slides: Regularity of the two-phase free boundaries.

Acronym:

Title: * Variational approach to the regularity of the free boundaries *

Duration:

Starting date:

Primary coordinator: * Bozhidar Velichkov *

Host institution: * Università di Pisa *

Funding Agency:

Funding Scheme:

Call year:

Reference:

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)"*

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,

and the team of Fostering Science for the support during the preparation of the proposal.