Variational approach to the regularity of the free boundaries

Project Information

VAREG

Grant agreement ID: 853404

DOI

10.3030/853404

Project closed

EC signature date 27 February 2020

Start date 1 June 2020

End date 30 November 2025

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 330 325,00

EU contribution

€ 1 330 325,00

Coordinated by

UNIVERSITA DI PISA
Italy

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Links to deliverables and publications from FP7 projects, as well as links to some specific result types such as dataset and software, are dynamically retrieved from OpenAIRE .

Publications

Peer reviewed articles (50)

Classical solutions to the soap film capillarity problem for plane boundaries

Author(s): Giulia Bevilacqua, Salvatore Stuvard, Bozhidar Velichkov
Published in: Mathematische Annalen, Issue 392, 2025, Page(s) 4607-4659, ISSN 0025-5831
Publisher: Springer Verlag
DOI: 10.1007/s00208-025-03172-z

A one-sided two phase Bernoulli free boundary problem

Author(s): Lorenzo Ferreri, Bozhidar Velichkov
Published in: Journal de Mathématiques Pures et Appliquées, Issue 195, 2025, Page(s) 103659, ISSN 0021-7824
Publisher: Elsevier BV
DOI: 10.1016/j.matpur.2025.103659

Regularity of shape optimizers for some spectral fractional problems

Author(s): Giorgio Tortone
Published in: Journal of Functional Analysis, Issue 1, 2021, ISSN 0022-1236
Publisher: Academic Press
DOI: 10.1016/j.jfa.2021.109271

Regularity of the optimal sets for the second Dirichlet eigenvalue

Author(s): Mazzoleni, Dario; Trey, Baptiste; Velichkov, Bozhidar
Published in: Annales de l'Institut Henri Poincaré C, Issue 39, 2022, Page(s) 529-573, ISSN 0294-1449
Publisher: Elsevier BV
DOI: 10.48550/arxiv.2010.00441

A Short Proof of Allard’s and Brakke’s Regularity Theorems

Author(s): Guido De Philippis, Carlo Gasparetto, Felix Schulze
Published in: International Mathematics Research Notices, Issue 2024, 2024, Page(s) 7594-7613, ISSN 1073-7928
Publisher: Oxford University Press
DOI: 10.1093/imrn/rnad281

Numerical computation of the cut locus via a variational approximation of the distance function

Author(s): François Générau, Edouard Oudet, Bozhidar Velichkov
Published in: ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 2022, ISSN 2822-7840
Publisher: EDP Sciences

Epsilon-regularity for the solutions of a free boundary system

Author(s): Francesco Paolo Maiale, Giorgio Tortone, Bozhidar Velichkov
Published in: Revista Matemática Iberoamericana, 2023, ISSN 0213-2230
Publisher: Universidad Autonoma de Madrid
DOI: 10.4171/rmi/1430

Regularity of the Optimal Sets for a Class of Integral Shape Functionals

Author(s): Giuseppe Buttazzo, Francesco Paolo Maiale, Dario Mazzoleni, Giorgio Tortone, Bozhidar Velichkov
Published in: Archive for Rational Mechanics and Analysis, Issue 248, 2024, ISSN 0003-9527
Publisher: Springer Verlag
DOI: 10.1007/s00205-024-01984-y

A survey on the optimal partition problem

Author(s): Roberto Ognibene, Bozhidar Velichkov
Published in: La Matematica, Issue 5, 2026, ISSN 2730-9657
Publisher: Springer Science and Business Media LLC
DOI: 10.1007/s44007-025-00176-8

On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity

Author(s): Nicola Soave, Giorgio Tortone
Published in: Archive for Rational Mechanics and Analysis, Issue 248, 2024, ISSN 0003-9527
Publisher: Springer Verlag
DOI: 10.1007/s00205-024-01970-4

Liouville theorems and optimal regularity in elliptic equations

Author(s): Giorgio Tortone
Published in: Proceedings of the London Mathematical Society, Issue 128, 2024, ISSN 0024-6115
Publisher: Oxford University Press
DOI: 10.1112/plms.12587

Free boundary cluster with Robin condition on the transmission interface

Author(s): Serena Guarino Lo Bianco, Domenico Angelo La Manna, Bozhidar Velichkov
Published in: Calculus of Variations and Partial Differential Equations, Issue 64, 2025, ISSN 0944-2669
Publisher: Springer Verlag
DOI: 10.1007/s00526-024-02864-9

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

Author(s): Lorenzo Ferreri, Bozhidar Velichkov
Published in: Transactions of the American Mathematical Society, Issue 377, 2024, Page(s) 7847-7876, ISSN 0002-9947
Publisher: American Mathematical Society
DOI: 10.1090/tran/9248

Quantitative spectral stability for the Neumann Laplacian in domains with small holes

Author(s): Veronica Felli, Lorenzo Liverani, Roberto Ognibene
Published in: Journal of Functional Analysis, Issue 288, 2025, Page(s) 110817, ISSN 0022-1236
Publisher: Academic Press
DOI: 10.1016/j.jfa.2024.110817

On the logarithmic epiperimetric inequality for the obstacle problem

Author(s): Luca Spolaor; Bozhidar Velichkov
Published in: Mathematics in Engineering, 2021, ISSN 2640-3501
Publisher: AIMS Press
DOI: 10.3934/mine.2021004

Sharp Behavior of Dirichlet–Laplacian Eigenvalues for a Class of Singularly Perturbed Problems

Author(s): Laura Abatangelo, Roberto Ognibene
Published in: SIAM Journal on Mathematical Analysis, Issue 56, 2024, Page(s) 474-500, ISSN 0036-1410
Publisher: Society for Industrial and Applied Mathematics
DOI: 10.1137/23m1564444

Eigenvalues of the Laplacian with moving mixed boundary conditions: The case of disappearing Neumann region

Author(s): Veronica Felli, Benedetta Noris, Roberto Ognibene
Published in: Journal of Differential Equations, Issue 320, 2025, Page(s) 247-315, ISSN 0022-0396
Publisher: Academic Press
DOI: 10.1016/j.jde.2022.02.052

A two-phase problem with Robin conditions on the free boundary

Author(s): Serena Guarino Lo Bianco; Domenico Angelo La Manna; Bozhidar Velichkov
Published in: Journal de l'Ecole polytechnique - Mathematiques, 2021, ISSN 2429-7100
Publisher: Ecole polytechnique
DOI: 10.5802/jep.139

The regularity problem in domains with lower dimensional boundaries

Author(s): Zanbing Dai; Joseph Feneuil; Svitlana Mayboroda
Published in: Journal of Functional Analysis, Issue 284, 2023, ISSN 0022-1236
Publisher: Academic Press
DOI: 10.1016/j.jfa.2023.109903

Green function estimates on complements of low-dimensional uniformly rectifiable sets

Author(s): Guy David, Joseph Feneuil, Svitlana Mayboroda
Published in: Mathematische Annalen, Issue 385, 2023, Page(s) 1797-1821, ISSN 0025-5831
Publisher: Springer Verlag
DOI: 10.1007/s00208-022-02379-8

Second-order estimates for the p-Laplacian in RCD spaces

Author(s): Luca Benatti, Ivan Yuri Violo
Published in: Journal of Differential Equations, Issue 439, 2025, Page(s) 113398, ISSN 0022-0396
Publisher: Academic Press
DOI: 10.1016/j.jde.2025.113398

"Non-rigidity of the absolutely continuous part of <mml:math xmlns:mml=""http://www.w3.org/1998/Math/MathML"" altimg=""si1.svg""> <mml:mi mathvariant=""script"">A</mml:mi> </mml:math> -free measures"

Author(s): Luigi De Masi, Carlo Gasparetto
Published in: Journal of Functional Analysis, Issue 289, 2025, Page(s) 111114, ISSN 0022-1236
Publisher: Academic Press
DOI: 10.1016/j.jfa.2025.111114

Effects of surface tension and elasticity on critical points of the Kirchhoff–Plateau problem

Author(s): Giulia Bevilacqua, Chiara Lonati
Published in: Bollettino dell'Unione Matematica Italiana, Issue 17, 2024, Page(s) 221-240, ISSN 1972-6724
Publisher: Unione Matematica Italiana
DOI: 10.1007/s40574-023-00392-6

A vectorial problem with thin free boundary

Author(s): Daniela De Silva, Giorgio Tortone
Published in: Calculus of Variations and Partial Differential Equations, Issue 62, 2023, ISSN 0944-2669
Publisher: Springer Verlag
DOI: 10.1007/s00526-023-02561-z

Soap films: from the Plateau problem to deformable boundaries

Author(s): Giulia Bevilacqua, Luca Lussardi, Alfredo Marzocchi
Published in: Communications in Applied and Industrial Mathematics, Issue 15, 2025, Page(s) 137-155, ISSN 2038-0909
Publisher: Walter de Gruyter GmbH
DOI: 10.2478/caim-2024-0019

Regularity of the optimal sets for the second Dirichlet eigenvalue

Author(s): Mazzoleni, Dario; Trey, Baptiste; Velichkov, Bozhidar
Published in: Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 2022, ISSN 0294-1449
Publisher: Elsevier BV
DOI: 10.4171/aihpc/14

An epiperimetric inequality for odd frequencies in the thin obstacle problem

Author(s): Matteo Carducci, Bozhidar Velichkov
Published in: Journal of Functional Analysis, Issue 289, 2025, Page(s) 111115, ISSN 0022-1236
Publisher: Academic Press
DOI: 10.1016/j.jfa.2025.111115

Green functions and smooth distances

Author(s): Joseph Feneuil, Linhan Li, Svitlana Mayboroda
Published in: Mathematische Annalen, Issue 389, 2024, Page(s) 2637-2727, ISSN 0025-5831
Publisher: Springer Verlag
DOI: 10.1007/s00208-023-02715-6

Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality

Author(s): François Générau; Édouard Oudet; Bozhidar Velichkov
Published in: Archive for Rational Mechanics and Analysis volume, Issue 246, 2022, Page(s) 561-602, ISSN 0003-9527
Publisher: Springer Verlag
DOI: 10.1007/s00205-022-01821-0

Elastic membranes spanning deformable curves

Author(s): Francesco Ballarin, Giulia Bevilacqua, Luca Lussardi, Alfredo Marzocchi
Published in: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Issue 104, 2024, ISSN 0044-2267
Publisher: John Wiley & Sons Ltd.
DOI: 10.1002/zamm.202300890

Local multiplicity for fractional linear equations with Hardy potentials

Author(s): Edoardo Mainini, Roberto Ognibene, Bruno Volzone
Published in: Calculus of Variations and Partial Differential Equations, Issue 64, 2025, ISSN 0944-2669
Publisher: Springer Verlag
DOI: 10.1007/s00526-024-02914-2

On the isoperimetric Riemannian Penrose inequality

Author(s): Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri
Published in: Communications on Pure and Applied Mathematics, 2024, ISSN 0010-3640
Publisher: John Wiley & Sons Inc.
DOI: 10.1002/cpa.22239

Quantitative spectral stability for Aharonov–Bohm operators with many coalescing poles

Author(s): Veronica Felli, Benedetta Noris, Roberto Ognibene, Giovanni Siclari
Published in: Journal of the European Mathematical Society, 2025, ISSN 1435-9855
Publisher: European Mathematical Society Publishing House
DOI: 10.4171/jems/1632

Regularity of the free boundary for the two-phase Bernoulli problem

Author(s): Guido De Philippis, Luca Spolaor, Bozhidar Velichkov
Published in: Inventiones Mathematicae, 2021, ISSN 1432-1297
Publisher: Springer
DOI: 10.1007/s00222-021-01031-7

Carleson perturbations for the regularity problem

Author(s): Zanbing Dai, Joseph Feneuil, Svitlana Mayboroda
Published in: Revista Matemática Iberoamericana, Issue 39, 2024, Page(s) 2119-2170, ISSN 0213-2230
Publisher: Universidad Autonoma de Madrid
DOI: 10.4171/rmi/1401

Numerical computation of the cut locus <i>via</i> a variational approximation of the distance function

Author(s): François Générau, Edouard Oudet, Bozhidar Velichkov
Published in: ESAIM: Mathematical Modelling and Numerical Analysis, Issue 56, 2022, Page(s) 105-120, ISSN 2822-7840
Publisher: EDP Sciences
DOI: 10.1051/m2an/2021088

Vectorial Bernoulli Problems and Free Boundary Systems

Author(s): Giorgio Tortone, Bozhidar Velichkov
Published in: La Matematica, Issue 5, 2026, ISSN 2730-9657
Publisher: Springer Nature
DOI: 10.1007/s44007-026-00195-z

A variational analysis of nematic axisymmetric films: the covariant derivative case

Author(s): G. Bevilacqua, C. Lonati, L. Lussardi, A. Marzocchi
Published in: Calculus of Variations and Partial Differential Equations, Issue 65, 2025, ISSN 0944-2669
Publisher: Springer Verlag
DOI: 10.1007/s00526-025-03182-4

Shape optimization problems in control form

Author(s): Giuseppe Buttazzo, Francesco Paolo Maiale, Bozhidar Velichkov
Published in: RENDICONTI LINCEI - MATEMATICA E APPLICAZIONI, 2021, ISSN 1120-6330
Publisher: Accademia Nazionale dei Lincei
DOI: 10.4171/rlm/942

Regularity of minimal surfaces with capillary boundary conditions

Author(s): Luigi De Masi, Nick Edelen, Carlo Gasparetto, Chao Li
Published in: Communications on Pure and Applied Mathematics, Issue 78, 2025, Page(s) 2436-2502, ISSN 0010-3640
Publisher: John Wiley & Sons Inc.
DOI: 10.1002/cpa.70008

Higher Order Boundary Harnack Principle via Degenerate Equations

Author(s): Susanna Terracini, Giorgio Tortone, Stefano Vita
Published in: Archive for Rational Mechanics and Analysis, Issue 248, 2024, ISSN 0003-9527
Publisher: Springer Verlag
DOI: 10.1007/s00205-024-01973-1

(Quasi-)conformal methods in two-dimensional free boundary problems

Author(s): Guido De Philippis, Luca Spolaor, Bozhidar Velichkov
Published in: Journal of the European Mathematical Society, 2024, ISSN 1435-9855
Publisher: European Mathematical Society Publishing House
DOI: 10.4171/jems/1435

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

Author(s): null null, Luca Benatti, null null, null null, Mattia Fogagnolo, null null, Lorenzo Mazzieri, null null
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2023, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2023.091

The boundary Harnack principle on optimal domains

Author(s): Francesco Paolo Maiale, Giorgio Tortone, Bozhidar Velichkov
Published in: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2022, ISSN 0391-173X
Publisher: Stampacchia Guido
DOI: 10.2422/2036-2145.202112_003

Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem

Author(s): Maria Colombo; Luca Spolaor; Bozhidar Velichkov
Published in: Interfaces and Free Boundaries, Issue 2, 2021, ISSN 1463-9963
Publisher: European Mathematical Society Publishing House
DOI: 10.4171/ifb/452

On the spectrum of sets made of cores and tubes

Author(s): Francesca Bianchi, Lorenzo Brasco, Roberto Ognibene
Published in: Journal of Convex Analysis, 2024, ISSN 0944-6532
Publisher: Heldermann Verlag

A strong maximum principle for minimizers of the one-phase Bernoulli problem

Author(s): Nick Edelen, Luca Spolaor, Bozhidar Velichkov
Published in: Indiana University Mathematics Journal, Issue 73, 2024, Page(s) 1061-1096, ISSN 0022-2518
Publisher: Indiana University Press
DOI: 10.1512/iumj.2024.73.9922

The symmetric (log-)epiperimetric inequality and a decay-growth estimate

Author(s): Nick Edelen, Luca Spolaor, Bozhidar Velichkov
Published in: Calculus of Variations and Partial Differential Equations, Issue 63, 2024, ISSN 0944-2669
Publisher: Springer Verlag
DOI: 10.1007/s00526-023-02610-7

On asymptotics of Robin eigenvalues in the Dirichlet limit

Author(s): Roberto Ognibene
Published in: Communications in Partial Differential Equations, Issue 50, 2025, Page(s) 1174-1210, ISSN 0360-5302
Publisher: Marcel Dekker Inc.
DOI: 10.1080/03605302.2025.2536098

Global Solutions for Two-Phase Complex Fluids with Quadratic Anchoring in Soft Matter Physics

Author(s): Giulia Bevilacqua, Andrea Giorgini
Published in: SIAM Journal on Mathematical Analysis, Issue 56, 2024, Page(s) 6057-6120, ISSN 0036-1410
Publisher: Society for Industrial and Applied Mathematics
DOI: 10.1137/23m1608902

Book chapters (2)

Geometric Invariants of Non-smooth Framed Curves

Author(s): Giulia Bevilacqua, Luca Lussardi, Alfredo Marzocchi
Published in: Springer INdAM Series, Anisotropic Isoperimetric Problems and Related Topics, 2024, Page(s) 1-13
Publisher: Springer Nature Singapore
DOI: 10.1007/978-981-97-6984-1_1

Isoperimetric Sets in Nonnegative Scalar Curvature and Their Role Through Various Concepts of Mass

Author(s): Luca Benatti, Mattia Fogagnolo
Published in: Springer INdAM Series, Anisotropic Isoperimetric Problems and Related Topics, 2024, Page(s) 169-199
Publisher: Springer Nature Singapore
DOI: 10.1007/978-981-97-6984-1_8

Monographic books (1)

Regularity of the One-phase Free Boundaries

Author(s): Bozhidar Velichkov
Published in: Lecture Notes of the Unione Matematica Italiana, 2023, ISBN 978-3-031-13237-7
Publisher: Springer International Publishing
DOI: 10.1007/978-3-031-13238-4