Francesco Maggi - Research


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Ars Inveniendi
Analytica 



Sets of Finite Perimeter &
Geometric Variational Problems

Optimal Mass Transport on Euclidean Spaces

In this page you can find a list of publications, divided by books and papers, a list of talks, with either slides or videos available, and a list of short reports from Oberwolfach meetings, which provide very concise and I think, helpful, accounts on some groups of papers.


Books and lecture notes:
  1. Maggi, Francesco. Optimal mass transport in Euclidean spaces, to appear in Cambridge Studies in Advanced Mathematics 207, Cambridge University Press, 2023.
  2. Maggi, Francesco. Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, 2012.
  3. Maggi, Francesco. Symmetrization, optimal transport and quantitative isoperimetric inequalities. This is a chapter in: Optimal transportation, Geometry and Functional inequalities (Edited by Luigi Ambrosio). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, 11. Edizioni della Normale, Pisa, 2010.
Publication list:
  1. Bonforte, Matteo; Maggi, Francesco; Restrepo, Daniel. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow. Preprint arXiv:2407.18868
  2. Maggi, Francesco; Novack, Michael; Restrepo, Daniel. A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation. Preprint arxiv:2312.11139
  3. Maggi, Francesco; Novack, Michael; Restrepo, Daniel. Plateau borders in soap films and Gauss' capillarity theory. Preprint arXiv:2310.20169.
  4. Fusco, Nicola; Maggi, Francesco; Morini, Massimiliano; Novack, Michael; Rigidity and large volume residues in exterior isoperimetry for convex sets. Preprint arXiv:2310.13569.
  5. Maggi, Francesco; Santilli, Mario. Rigidity and compactness with constant mean curvature in warped product manifolds, Preprint arXiv:2303.03499
  6. Maggi Francesco; Neumayer, Robin; Tomasetti, Ignacio. Rigidity theorems for best Sobolev inequalities. Preprint arXiv:2206.12386. Accepted on Advances in Mathematics, 2023.
  7. Maggi, Francesco; Novack, Michael. Isoperimetric residues and a mesoscale flatness criterion for hypersurfaces with bounded mean curvature. Preprint arXiv:2205.02951
  8. Maggi, Francesco; Restrepo, Daniel. Uniform stability in the Euclidean isoperimetric problem for the Allen--Cahn energy. Preprint arXiv:2202.11583. Accepted on Analysis & PDE, 2022.
  9. Dipierro, Serena; Maggi, Francesco; Valdinoci, Enrico. Minimizing cones for fractional capillarity problems. Preprint arXiv:2008.06175 Accepted on Revista Matematica Iberoamericana 2021.
  10. King, Darren; Maggi, Francesco; Stuvard, Salvatore. Smoothness of collapsed regions in a capillarity model for soap films. Preprint arXiv:2007:14868 Accepted on Archive for Rational Mechanics and Analysis, 2021.
  11. Bernstein, Jacob; Maggi, Francesco; Symmetry and rigidity of minimal surfaces with Plateau-type singularities. Preprint arXiv:2003.01784. Accepted on Archive for Rational Mechanics and Analysis, 2020.
  12. King, Darren; Maggi, Francesco; Stuvard, Salvatore. Collapsing and the convex hull property in a soap film capillarity model. Preprint arXiv:2002.06273
    Accepted on  Ann. Inst. Henri Poincaré C, Analyse Non Linéaire, 2021.
  13. King, Darren; Maggi, Francesco; Stuvard, Salvatore. Plateau's problems as as singular limit of capillarity problems. Accepted on Comm. Pure Applied Math. Preprint arXiv:1907.00551
  14. Maggi, Francesco; Scardicchio, Antonello; Stuvard, Salvatore. Soap films with gravity and almost-minimal surfaces. Accepted on Disc. Cont. Dynamical Systems. Volume in honor of Luis Caffarelli's 70th birthday. Preprint arXiv:1807.05200
  15. Delgadino, Matias; Maggi, Francesco. Alexandrov's theorem revisited. Accepted on Anal. PDE. Preprint arXiv:1711.07690
  16. Cavalletti, Fabio; Maggi, Francesco; Mondino, Andrea. Quantitative isoperimetry à la Lévy-Gromov. Accepted on Comm. Pure Appl. Math. Preprint arXiv:1707.04326
  17. Delgadino, Matias G.; Maggi, Francesco; Mihaila, Cornelia; Neumayer, Robin. Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals. (2018) Arch. Ration. Mech. Anal. 230, no. 3, 1131–1177. Preprint arXiv:1705.10117
  18. Cavalletti, Fabio; Maggi, Francesco; Mondino, Andrea; Rigidity for critical points in the Lévy-Gromov inequality. (2018) Math. Z. 289, no. 3-4, 1191–1197. Preprint arXiv:1612.04119
  19. Figalli, Alessio; Maggi, Francesco; Mooney, Connor. The sharp quantitative Euclidean concentration inequality. (2018) Camb. J. Math. 6, no. 1, 59–87. Preprint arXiv:1601.04100v3
  20. Dipierro, Serena; Maggi, Francesco; Valdinoci, Enrico. Asymptotic expansions of the contact angle in nonlocal capillarity problems. (2017) J. Nonlinear Sci. 27 , no. 5, 1531–1550. Preprint arXiv:1610.00075
  21. Maggi, Francesco; Valdinoci, Enrico. Capillarity problems with nonlocal surface tension energies. (2017) Comm. Partial Differential Equations 42, no. 9, 1403–1446. Preprint arXiv:1606.08610
  22. Ciraolo, Giulio; Figalli, Alessio; Maggi, Francesco. A quantitative analysis of metrics in R^n with almost constant positive scalar curvature, with applications to fast diffusion flows. (2017) International Mathematics Research Notices, rnx071, Preprint arXiv:1602.01916
  23. Cicalese, Marco; Leonardi, Gian Paolo; Maggi, Francesco Sharp stability inequalities for planar double bubbles. (2017) Interfaces Free Bound. 19, no. 3, 305–350. Preprint arXiv:1211.3698
  24. Ciraolo, Giulio; Figalli, Alessio; Maggi, Francesco; Novaga, Matteo; Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. (2018) J. Reine Angew. Math. 741, 275–294. Preprint arXiv:1503.00653
  25. Carlen, Eric; Maggi, Francesco. Stability for the Brunn-Minkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry. (2017) Canad. J. Math. 69, 1036-1063. Preprint arXiv:1507.03454.
  26. Maggi, Francesco; Neumayer, Robin. A bridge between Sobolev and Escobar inequalities and beyond (2017) J. Funct. Anal. 273(6), 2070-2106. Preprint arXiv:1609.02346
  27. Krummel, Brian; Maggi, Francesco. Isoperimetry with upper mean curvature bounds and sharp stability estimates (2017) Calc. Var. PDE. 56(2), Paper no. 53, 43 pp. Preprint arXiv:1606.00490
  28. Ciraolo, Giulio; Maggi, Francesco. On the shape of compact hypersurfaces with almost constant mean curvature (2017) Comm. Pure Appl. Math. 70(4), 665-716. Preprint arXiv:1503.06674
  29. Colombo, Maria; Maggi, Francesco. Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters (2017) Nonlinear Anal. 153, 243-274. Preprint arXiv:1605.05641
  30. Leonardi, Gian Paolo; Maggi, Francesco (2017)  Improved convergence theorems for bubble clusters. II. The three-dimensional case. Indiana Univ. Math. J. 66(2), 559-608. Preprint arXiv:1505.06709.
  31. De Lellis, Camillo; Ghiraldin, Francesco; Maggi, Francesco (2017) A direct approach to Plateau's problem. J. Eur. Math. Soc. (JEMS) 19(8), 2219-2240. Preprint arXiv:1408.4047
  32. Cagnetti, Filippo; Colombo, Maria; De Philippis, Guido; Maggi Francesco (2017) Essential connectedness and the rigidity problem for Gaussian symmetrization. J. Eur. Math. Soc. (JEMS) 19(2) 395-439. Preprint arXiv:1304.4527
  33. De Philippis, Guido; Maggi, Francesco (2017). Dimensional estimates for singular sets in geometric variational problems with free boundaries. J. Reine Angew. Math. (Crelle's Journal), 725, 217-234. Preprint arXiv:1407.4834
  34. Maggi, Francesco; Mihaila, Cornelia. On the shape of capillarity droplets in a container (2016) Calc. Var. PDE. 55(5), Paper no. 122, 42 pp. Preprint arXiv:1509.03324v1
  35. Cicalese, Marco; Leonardi, Gian Paolo; Maggi, Francesco. (2016) Improved convergence theorems for bubble clusters. I. The planar case. Indiana Univ. Math. J. 65(6), 1979-2050. Preprint arXiv:1409.6652.
  36. Caroccia, Marco; Maggi, Francesco. (2016) A sharp quantitative version of Hales' isoperimetric honeycomb theorem, J. Math. Pures Appl. (9) 106(5), 935-956. Preprint arXiv:1410.6128.
  37. Figalli, Alessio; Fusco, Nicola; Maggi, Francesco; Millot, Vincent; Morini, Massimiliano (2015). Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys. 336(1), 441-507. Preprint arXiv:1403.0516
  38. De Philippis, Guido; Maggi, Francesco (2015). Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law. Arch. Ration. Mech. Anal. 216(2), 473-568. Preprint arXiv:1402.0549
  39. Cagnetti, Filippo; Colombo, Maria; De Philippis, Guido; Maggi Francesco (2014). Rigidity of equality cases in Steiner's perimeter inequality. Anal. PDE, 7(7), 1535-1593. Preprint arXiv:1309.1639
  40. De Philippis, Guido; Maggi, Francesco (2014). Sharp stability inequalities for the Plateau problem. J. Differential Geom. 96(3), 399-456.
  41. Maggi, Francesco; Ponsiglione, Marcello; Pratelli, Aldo (2014) Quantitative stability in the isodiametric inequality via the isoperimetric inequality. Trans. AMS 366(3), 1141-1160.
  42. Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo (2014). A geometric approach to correlation inequalities in the plane. Ann. Inst. Henri Poincaré Probab. Stat. 50(1), 1-14.
  43. Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo (2013). Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math. 242, 80-101.
  44. Figalli, Alessio; Maggi, Francesco (2013) On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations 48(3-4), 447-489.
  45. Figalli, Alessio; Maggi, Francesco; (2011) On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201(1), 143-207.
  46. Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2011). On the isoperimetric problem with respect to a mixed Euclidean-Gaussian density. J. Funct. Anal. 260(12), 3678-3717.
  47. Cianchi, Andrea; Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2011) On the isoperimetric deficit in Gauss space. Amer. J. Math. 133(1), 131-186.
  48. Fonseca, Irene; Leoni, Giovanni; Maggi, Francesco; Morini, Massimiliano (2010) Exact reconstruction of color images by a total variation model, Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 1291-1331.
  49. Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo (2010). A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182, 167-211.
  50. Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo (2009). A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 2511-2519.
  51. Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo (2009). A note on Cheeger sets, Proc. AMS 137(6), 2057–2062.
  52. Cianchi, Andrea; Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2009), The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (5), 1105–1139.
  53. Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2009) Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 51–71.
  54. Maggi, Francesco (2008). Some methods for studying stability in isoperimetric type problems, Bull. AMS 45(3), 367-408.
  55. Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2008). The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168(3), 941-980.
  56. Maggi, Francesco; Villani, Cédric (2008). Balls have the worst best Sobolev inequalities. Part two: variants and extensions, Calc. Var. PDE 31(1), 47-74.
  57. Conti, Sergio; Maggi, Francesco, (2008). Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187(1), 1-48.
  58. Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo (2007) The sharp quantitative Sobolev inequality for functions of bounded variation J. Funct. Anal. 244(1) 315-341.
  59. Conti, Sergio; Maggi, Francesco; Müller, Stefan (2006) Rigorous derivation of Föppl’s theory for clamped elastic membranes leads to relaxation, SIAM J. Math. Anal. 38(2) 657-680.
  60. Fusco, Nicola; Gori, Michele; Maggi, Francesco (2006). A remark on Serrin’s theorem. NoDEA 13(4), 425-433.
  61. Conti, Sergio; Faraco, Daniel; Maggi, Francesco; Müller, Stefan (2005). Rank-one convex functions on 2 × 2 symmetric matrices and laminates on rank-three lines. Calc. Var. PDE 24(4), 479-493.
  62. Conti, Sergio; Faraco, Daniel; Maggi, Francesco (2005) A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175(2), 287-300.
  63. Gori, Michele; Maggi, Francesco (2005). The common root of the geometric conditions in Serrin’s lower semicontinuity theorem. Ann. Mat. Pura e Applicata,184(1), 95-114.
  64. Maggi, Francesco; Villani, Cédric (2005). Balls have the worst best Sobolev inequalities. J. Geom. Anal. 15(1), 83-121.
  65. Maggi, Francesco; Morini, Massimiliano (2004). A Γ-convergence result for variational integrators of quadratic lagrangians. ESAIM: COCV 10(4), 656-665.
  66. Maggi, Francesco (2003) On the relaxation on BV of certain non-coercive integral functionals, J. Convex Anal. 10(2), 477-489.
  67. Gori, Michele; Maggi, Francesco (2003) On the lower semicontinuity of supremal functionals, ESAIM: COCV 9, 135-143.
  68. Gori, Michele; Maggi, Francesco; Marcellini, Paolo (2003). On some sharp conditions for lower semicontinuity in L1. Diff. Int. Equations 16(1), 51-76.
Oberwolfach reports:


Oberwolfach reports usually provide nice extended abstracts of one or more connected papers. I have collected here the Oberwolfach reports covering my work, and the webpages of the corresponding workshops (the full set of reports for each workshop can be downloaded on these webpages).
  1. Report (written by Aldo Pratelli and myself) on the sharp Euclidean isoperimetric inequality -- Calculus of Variations, 2006
  2. Report (by Alessio Figalli) on the stability of small crystals under an external potential -- Partielle Differentialgleichungen, 2009
  3. Report (by Nicola Fusco) on the stability of Gaussian isoperimetric inequalities -- Calculus of Variations, 2010
  4. Report on rigidity for symmetrization inequalities in Euclidean and Gaussian geometry -- Partial Differential Equations, 2013
  5. Report on regularity theory and dimensional estimates for anisotropic capillarity problems -- Calculus of Variations, 2014
  6. Report on stability for Euclidean concentration and applications to Statistical Mechanics -- Calculus of Variations, 2016
  7. Report (by Brian Krummel) on the stability of Almgren's isoperimetric principle -- Calculus of Variations, 2016
  8. Report (by Fabio Cavalletti) on a quantitative version of the Levy-Gromov isoperimetric comparison theorem -- Partial Differential Equations, 2017
  9. Report on critical and almost-critical points in isoperimetric problems -- Calculus of Variations, 2018
  10. Report on a capillarity model for soap films -- Partial Differential Equations, 2019
  11. Report on a characterization of catenoids among minimal Plateau's surfaces -- Partial Differential Equations, 2021
 Slides and videos:
  1. A mesoscale flatness criterion and its application to exterior isoperimetry, Roscoff, France, June 22 (slides)
  2. A model for soap films based on capillarity theory, One World PDE seminar, May 20 (video, slides);
  3. Plateau's problem as a capillarity problem, IAS Princeton, February 19 (video);
  4. Isoperimetry and boundaries with almost constant mean curvature, IAS Princeton February 19 (video);
  5. Alexandov's theorem revisited, Brown U and Rice U, Spring 18;
  6. Compactness of critical points for elliptic isoperimetric problems, Aggregation-diffusion PDEs: Variational principles, nonlocalities, and systems, Anacapri, 7/17;
  7. Isoperimetric theorems, open problems and new results, International Centre for Theoretical Physics Colloquium, 2/17 (video);
  8. Boundaries with almost-constant mean curvature, Nonlinear Analysis Seminar, Rutgers U, 2/17;
  9. Stability theorems for geometric variational problems, Colloquium talk, Courant Institute NY, 2/17;
  10. Almost critical points in geometric variational problems: the Euclidean isoperimetric problem, James Serrin: from his legacy to the new frontiers, Perugia, 1/17;
  11. Quantitative isoperimetric principles and applications to phase transitions, Calc Var and Nonlinear PDE Conference 2016, Columbia U;
  12. Almost constant mean curvature hypersurfaces and capillarity theory. SIAM PDE Conference 2015, plenary talk, 12/15 (video);
  13. Improved convergence to singular minimizers and applications. SIAM PDE Conference 2015, invited talk, 12/15 (slides);
  14. A quantitative description of almost constant mean curvature hypersurfaces. Institut Fourier, Grenoble, France, 7/15. (video);
  15. A general compactness theorem for Plateau's problem. Conference on Calculus of Variations. Fields Institute, Toronto, Canada, 11/14 (video);
  16. Regularity of free boundaries in anisotropic variational problems and the validity of Young's law. Analysis of PDE: Theory, methods and applications, Protaras, Cyprus, 7/14 (slides).