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.

    • 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


Filippo Paiano (PhD student at University of Pisa)

Matteo Carducci (PhD student at Scuola Normale Superiore)

Carlo Gasparetto (post-doc starting on 1 April 2023)

Luca Benatti (post-doc since 1 February 2023)

Lorenzo Ferreri (PhD student at Scuola Normale Superiore)

Giulia Bevilacqua (post-doc since 1 September 2022)

Roberto Ognibene (post-doc since 1 February 2022)

Joseph Feneuil (post-doc from 1 September 2021 to 9 July 2022)

Giorgio Tortone (post-doc since 1 March 2021)

Bozhidar Velichkov (PI)


Workshop Calculus of Variations and Free Boundary Problems VI (2023)

Variational methods for thin structures and free boundary problems (2023)

Regularity Theory for Free Boundary and Geometric Variational Problems III (2023)

Workshop Calculus of Variations and Free Boundary Problems V (2023)

Workshop Calculus of Variations and Free Boundary Problems IV (2022)

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

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


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.

18/05/2023. Luca Spolaor (UC San Diego).
Title: Interior regularity for stationary two-dimensional multivalued maps.

Abstract. Q-valued maps minimizing a suitably defined Dirichlet energy were introduced by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren's results to stationary Q-valued maps in dimension 2.
This is joint work with Jonas Hirsch (Leipzig).

05/04/2023. Roberto Ognibene (Università di Pisa).
Title: Capacity and torsional rigidity: two measures of spectral stability.

Abstract. In this talk, I will discuss the behavior of the spectrum of the Laplacian on bounded domains, subject to varying mixed boundary conditions. More precisely, let us assume the boundary of the domain to be split into two parts, on which homogeneous Neumann and Dirichlet boundary conditions are respectively prescribed; let us then assume that, alternately, one of these regions “disappears” and the other one tends to cover the whole boundary. In this framework, I will first describe under which conditions the eigenvalues of the mixed problem converge to the ones of the limit problem (where a single kind of boundary condition is imposed); then, I will sharply quantify the rate of this convergence by providing an explicit first-order asymptotic expansion of the “perturbed” eigenvalues. These results have been obtained in collaboration with L. Abatangelo, V. Felli and B. Noris.

14/03/2023 (cancelled). Carlos Kenig (University of Chicago).
New channels of energy for wave equations, new non-radiative solutions and soliton resolution.

Abstract. We will discuss the role of nonradiative solutions to nonlinear wave equations, in connection with soliton resolution. Using new channels of energy estimates we characterize all radial nonradiative solutions for a general class of nonlinear wave equations. This is joint work with C.Collot, T. Duyckaerts, and F. Merle.

08/03/2023. Damià Torres-Latorre (Universitat de Barcelona).
Title: Optimal regularity for supercritical parabolic obstacle problems.

Abstract. The parabolic nonlocal obstacle problem is said to be in the supercritical regime (s < 1/2) when the time derivative is of higher order than the diffusion operator. We will discuss the optimal C1,1 regularity of solutions and the C1,α regularity of the free boundary. The arguments rely on comparison principles and the scaling of the equation to circumvent the fact that blow-ups, the usual technique for free boundary problems, are not useful in this context. This is a joint work with X. Ros-Oton.

01/03/2023. Seongmin Jeon (KTH).
Almost minimizers for the parabolic thin obstacle problem.

Abstract. We consider almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their Hσ, σ/2 - regularity for every σ, strictly between zero and one, as well as Hβ,β/2 - regularity of their spatial gradients on the either side of the thin space for some β again in (0,1). We then extend these regularity results to the variable Hölder continuous coefficient setting. We also discuss the regularity of the "regular" part of the free boundary. This is based on joint work with Arshak Petrosyan.

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.
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.


25/09/2023 - 28/09/2023. Luigi De Masi (Università di Padova)

25/04/2023 - 27/05/2023. Luca Spolaor (UC San Diego)

25/02/2023 - 04/03/2023. Seongmin Jeon (KTH)

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)


F. Generau, E. Oudet, B. Velichkov. Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality. Arch. Rat. Mech. Anal. (2022).

F. P. Maiale, G. Tortone, B. Velichkov. The boundary Harnack principle on optimal domains. Annali della Scuola Normale Superiore Cl. Sci. (to appear).

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. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (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).
          - Summary: Cover letter.

G. De Philippis, L. Spolaor, B. Velichkov.  Regularity of the free boundary for the two-phase Bernoulli problem. Invent. Math. (2021).
          - 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.