Variational approach to the regularity of the free boundaries

2020 - 2025

ERC Starting Grant 2019

      NEWS Team Workgroup Conferences Publications Project ID


      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.

    • Epiperimetric inequalities for the one-phase problem:
                                                            the 2D case, applications; singularities.

    • The logarithmic epiperimetric inequality: slides, obstacle, thin-obstacle.


PhD in Mathematics in Pisa


Giulia Bevilacqua (post-doc - starting date tba)

Roberto Ognibene (post-doc since 1 February 2022)

Joseph Feneuil (post-doc since 1 September 2021).

Giorgio Tortone (post-doc since 1 March 2021)

Bozhidar Velichkov (PI)


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.

25/05/2022. Salvatore Stuvard (Università degli Studi di Milano).
Existence of canonical multi-phase Brakke flows.
Abstract (pdf).

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.
Abstract (pdf).

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.
Abstract (pdf).

Invited speakers and seminars (before 2022)

25/10/2021. Alessandro Audrito (ETH Zürich).
A variational approach to a class of nonlinear Cauchy-Neumann problems.

9/7/2021. Mickaël Nahon (Université de Savoie).
Regularity of a free boundary transmission problem with obstacle.

11/11/2020. Giorgio Tortone (Università di Bologna) (online).
Vectorial problems with thin free boundaries.

21/10/2020. Susanna Terracini (Università degli Studi di Torino) (online).
Liouville type theorems and local behaviour of solutions to degenerate or singular problems.


Regularity Theory for Free Boundary and Geometric Variational Problems II (2022)

Regularity Theory for Free Boundary and Geometric Variational Problems (2021)


F. P. Maiale, G. Tortone, B. Velichkov. The boundary Harnack principle on optimal domains. Preprint 2021.

G. De Philippis, L. Spolaor, B. Velichkov. (Quasi-)conformal methods in two-dimensional free boundary problems. Preprint 2021.

F. P. Maiale, G. Tortone, B. Velichkov. Epsilon-regularity for the solutions of a free boundary system. Preprint 2021.

G. De Philippis, M. Engelstein, L. Spolaor, B. Velichkov. Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries. Preprint 2021.

G. Buttazzo, F. P. Maiale, B. Velichkov. Shape optimization problems in control form. Preprint 2021.

D. Mazzoleni, B. Trey, B. Velichkov. Regularity of the optimal sets for the second Dirichlet eigenvalue. Ann. Inst. H. Poincaré Anal. Non Linéaire (2022).
          Related documents:
          - Summary: Cover letter.

G. De Philippis, L. Spolaor, B. Velichkov.  Regularity of the free boundary for the two-phase Bernoulli problem. Invent. Math. (2021).
         Related documents:
          - Summary: Cover letter.
          - Oberwolfach report 2020: Regularity of the two-phase free boundaries.
          - Slides: Regularity of the two-phase free boundaries.

S. Guarino Lo Bianco, D. A. La Manna, B. Velichkov. A two-phase problem with Robin conditions on the free boundary. Journal de l'École polytechnique (2021).

Project ID

Acronym: VAREG

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