Variational approach to the regularity of the free boundaries

2019 - 2024

ERC Starting Grant 2019

                NEWS Team Events Publications Funding and 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.


New post-doc position

New post-doc position on the project VAREG is available at the University of Pisa.

Deadline for applications: 21 May 2021.

The applications should be submitted on the platform PICA.
Instructions are available on
the site of the call

Selected candidas will be invited to an (online) interview.
Interview date: 21 May 2021.

Starting date: 1 September 2021.


Giorgio Tortone (post-doc since 1 March 2021)

Bozhidar Velichkov (PI)



Regularity Theory for Free Boundary and Geometric Variational Problems.
Workshop scheduled for September 2021.


• D. Mazzoleni, B. Trey, B. Velichkov. Regularity of the optimal sets for the second Dirichlet eigenvalue. Preprint.
          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.