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A new glance to the Alt-Caffarelli-Friedman monotonicity formula

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: https://www.aimspress.com/newsinfo/1429.html

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in [18].
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Keywords monotonicity formulas; Heisenberg group; free boundary problems; two-phase problems

Citation: Fausto Ferrari, Nicolò Forcillo. A new glance to the Alt-Caffarelli-Friedman monotonicity formula. Mathematics in Engineering, 2020, 2(4): 657-679. doi: 10.3934/mine.2020030


  • 1. Alt W, Caffarelli L, Friedman A (1984) Variational problems with two phases and their free boundaries. T Am Math Soc 282: 431–461.    
  • 2. Athanasopoulos I, Caffarelli L, Salsa S (1996) Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems. Ann Math 143: 413–434.    
  • 3. Bonfiglioli A, Lanconelli E, Uguzzoni F (2007) Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Berlin: Springer.
  • 4. Birindelli I (2003) Superharmonic functions in the Heisenberg group: estimates and Liouville theorems. NODEA–Nonlinear Diff 10: 171–185.
  • 5. Brascamp HJ, Lieb EH (1976) On extensions of the Brunn-Minkowski and Prèkopa-Leindler Theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J Funct Anal 22: 366–389.    
  • 6. Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C1,α. Rev Mat Iberoamericana 3: 139–162.
  • 7. Caffarelli LA (1988) A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X. Ann Scuola Norm Sci 15: 583–602.
  • 8. Caffarelli LA, Jerison D, Kenig CE (2002) Some new monotonicity theorems with applications to free boundary problems. Ann Math 155: 369–404.    
  • 9. Caffarelli L, Salsa S (2005) A Geometric Approach to Free Boundary Problems, Providence RI: American Mathematical Society.
  • 10. Capogna L, Danielli D, Garofalo N (1994) The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Commun Anal Geom 22: 203–215.
  • 11. Capogna L, Danielli D, Pauls S, et al. (2007) An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Basel: Birkhauser Verlag.
  • 12. Danielli D, Garofalo N, Petrosyan A (2007)The sub-elliptic obstacle problem: C1,a regularity of the free boundary in Carnot groups of step two. Adv Math 211: 485–516.
  • 13. Danielli D, Garofalo N, Salsa S (2003) Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary. Indiana U Math J 52: 361–398.
  • 14. De Silva D, Ferrari F, Salsa S (2014) Two-phase problems with distributed sources: regularity of the free boundary. Anal PDE 7: 267–310.    
  • 15. Dipierro S, Karakhanyan AL (2018) A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two. Commun Part Diff Eq 43: 1073–1101.    
  • 16. Dzhugan A, Ferrari F (2020) Domain variation solutions for degenerate elliptic operators. arXiv:2001.07174.
  • 17. Folland GB (1975) Subelliptic estimates and function spaces on nilpotent Lie groups. Ark Mat 13: 161–207.    
  • 18. Ferrari F, Forcillo N (2020) Some remarks about the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group. arXiv:2001.04393.
  • 19. Ferrari F, Salsa S (2010) Regularity of the solutions for parabolic two-phase free boundary problems. Commun Part Diff Eq 35: 1095–1129.    
  • 20. Ferrari F, Valdinoci E (2011) Density estimates for a fluid jet model in the Heisenberg group. J Math Anal Appl 382: 448–468.    
  • 21. Franchi B, Serapioni R, Cassano FS (1996) Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J Math 22: 859–890.
  • 22. Franchi B, Serapioni R, Cassano FS (2003) On the structure of finite perimeter sets in step 2 Carnot groups. J Geom Anal 13: 421–466.    
  • 23. Franchi B, Serapioni R, Cassano FS (2003) Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups. Commun Anal Geom 11: 909–944.    
  • 24. Friedland S, Hayman WK (1976) Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment Math Helv 51: 133–161.    
  • 25. Garofalo N, Nhieu DM (1996) Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun Pure Appl Math 49: 1081–1144.    
  • 26. Garofalo N, Lanconelli E (1990) Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann I Fourier 40: 313–356.    
  • 27. Gilbarg D, Trudinger NS (2001) Elliptic Partial Differential Equations of Second Order Classics in Mathematics, Berlin: Springer-Verlag.
  • 28. Garofalo N, Rotz K (2015) Properties of a frequency of Almgren type for harmonic functions in Carnot groups. Calc Var Partial Dif 54: 2197–2238.    
  • 29. Greiner PC (1980) Spherical harmonics on the Heisenberg group. Can Math Bull 23: 383–396.    
  • 30. Hayman WK, Ortiz EL (1976) An upper bound for the largest zero of Hermite's function with applications to subharmonic functions. P Roy Soc Edinb A 75: 182–197.
  • 31. Jerison DS (1981) The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II. J Funct Anal 43: 224–257.    
  • 32. Magnani V (2001) Differentiability and area formula on stratified Lie groups. Houston J Math 27: 297–323.
  • 33. Matevosyan N, Petrosyan A (2011) Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Commun Pure Appl Math 64: 271–311.    
  • 34. Noris B, Tavares H, Terracini S, et al. (2010) Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun Pure Appl Math 63: 267–302.    
  • 35. Quitalo V (2013) A free boundary problem arising from segregation of populations with high competition. Arch Ration Mech Anal 210: 857–908.    
  • 36. Sperner jr E (1973) Zur symmetrisierung von funktionen auf sphären. Math Z 134: 317–327.    
  • 37. Terracini S, Tortone G, Vita S (2018) On s-harmonic functions on cones. Anal PDE 11: 1653– 1691.    
  • 38. Terracini S, Verzini G, Zilio A (2016) Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian. J Eur Math Soc 18: 2865–2924.    
  • 39. Teixeira EV, Zhang L (2011) Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds. Adv Math 226: 1259–1284.    


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