Skip to main content
SpringerLink
Log in

Geometric Analysis of Fractional Phase Transition Interfaces

  • Chapter
Book cover Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 2))

Abstract

We discuss some recent results on phase transition models driven by nonlocal operators, also in relation with their limit (either local or nonlocal) interfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The reader has noticed that this is the same threshold for the known results discussed in Sect. 3.1 when n=3. On the other hand, while the threshold s=1/2 is optimal here, the optimality for the results of Sect. 3.1 is completely open. Further regularity results for the limit interface when s→(1/2)− will be discussed in Sect. 3.5.

  2. 2.

    As usual, χ E denotes the characteristic function of the set E.

References

  1. Alberti, G., Bellettini, G.: A nonlocal anisotropic model for phase transitions, I. The optimal profile problem. Math. Ann. 310(3), 527–560 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alberti, G., Bouchitté, G., Seppecher, P.: Un résultat de perturbations singulières avec la norme H 1/2. C. R. Acad. Sci. Paris 319(I), 333–338 (1994)

    MATH  Google Scholar 

  3. Alberti, G., Bellettini, G., Cassandro, M., Presutti, E.: Surface tension in Ising systems with Kac potentials. J. Stat. Phys. 82, 743–796 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65(1–3), 9–33 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)

    Article  MATH  Google Scholar 

  6. Ambrosio, L., de Philippis, G., Martinazzi, L.: Γ-convergence of nonlocal perimeter functionals. Manuscr. Math. 134, 377–403 (2011)

    Article  MATH  Google Scholar 

  7. Barlow, M.T., Bass, R.F., Gui, C.: The Liouville property and a conjecture of De Giorgi. Commun. Pure Appl. Math. 53(8), 1007–1038 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barrios Barrera, B., Figalli, A., Valdinoci, E.: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2012, accepted for publication)

    Google Scholar 

  9. Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) 25(1–2), 69–94 (1997–1998) (Dedicated to Ennio De Giorgi)

    MathSciNet  Google Scholar 

  10. Berestycki, H., Hamel, F., Monneau, R.: One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103(3), 375–396 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brezis, H.: How to recognize constant functions. Connections with Sobolev spaces. Usp. Mat. Nauk 57(346), 59–74 (2002)

    Article  MathSciNet  Google Scholar 

  13. Cabré, X., Cinti, E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst. 28(3), 1179–1206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cabré, X., Cinti, E.: Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. Partial Differ. Equ. (2012, accepted for publication)

    Google Scholar 

  15. Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Preprint. http://arxiv.org/abs/1111.0796

  16. Cabré, X., Solà-Morales, J.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58(12), 1678–1732 (2005)

    Article  MATH  Google Scholar 

  17. Caffarelli, L., Córdoba, A.: Uniform convergence of a singular perturbation problem. Commun. Pure Appl. Math. 48, 1–12 (1995)

    Article  MATH  Google Scholar 

  18. Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41(1–2), 203–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Caffarelli, L., Valdinoci, E.: Regularity properties of nonlocal minimal surfaces via limiting arguments. Preprint, http://www.ma.utexas.edu/mp_arc/c/11/11-69.pdf

  20. Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Caffarelli, L., Roquejoffre, J.M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63(9), 1111–1144 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Caffarelli, L., Roquejoffre, J.M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12(5), 1151–1179 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Caputo, M.C., Guillen, N.: Regularity for non-local almost minimal boundaries and applications. Preprint. http://arxiv.org/pdf/1003.2470.pdf

  24. Craig, W., Groves, M.D.: Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19(4), 367–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Craig, W., Schanz, U., Sulem, C.: The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5(14), 615–667 (1997)

    Article  MathSciNet  Google Scholar 

  26. De Giorgi, E.: Convergence problems for functionals and operators. In: Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, pp. 131–188. Pitagora, Bologna (1979)

    Google Scholar 

  27. del Pino, M., Kowalczyk, M., Wei, J.: A counterexample to a conjecture by De Giorgi in large dimensions. C. R. Math. 346, 23–24 (2008)

    Google Scholar 

  28. del Pino, M., Kowalczyk, M., Wei, J.: Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. Rev. Unión Mat. Argent. 50(2), 95–107 (2009)

    MATH  Google Scholar 

  29. del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension N≥9. Ann. Math. 174, 1485–1569 (2011)

    Article  MATH  Google Scholar 

  30. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker.s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012). doi:10.1016/j.bulsci.2011.12.004. http://www.sciencedirect.com/science/article/pii/S0007449711001254

    Article  MathSciNet  MATH  Google Scholar 

  31. Dipierro, S., Figalli, A., Palatucci, G., Valdinoci, E.: Asymptotics of the s-perimeter as s↘0. Discrete Contin. Dyn. Syst., Ser. A (2012, accepted for publication)

    Google Scholar 

  32. Farina, A.: Symmetry for solutions of semilinear elliptic equations in ℝN and related conjectures. Ric. Mat. 48, 129–154 (1999)

    MathSciNet  MATH  Google Scholar 

  33. Farina, A., Valdinoci, E.: The state of art for a conjecture of De Giorgi and related questions. In: Reaction-Diffusion Systems and Viscosity Solutions. World Scientific, Singapore (2008)

    Google Scholar 

  34. Farina, A., Valdinoci, E.: Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications. Indiana Univ. Math. J. 60, 121–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Federer, H.: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Am. Math. Soc. 76, 767–771 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fefferman, C., de la Llave, R.: Relativistic stability of matter, I. Rev. Mat. Iberoam. 2(1–2), 119–213 (1986)

    Article  Google Scholar 

  37. Garroni, A., Müller, S.: Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36, 1943–1964 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Garroni, A., Müller, S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181, 535–578 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Garroni, A., Palatucci, G.: A singular perturbation result with a fractional norm. In: Variational Problems in Material Science. Progr. Nonlinear Differential Equations Appl., vol. 68, pp. 111–126. Birkhäuser, Basel (2006)

    Chapter  Google Scholar 

  40. Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. González, M.D.M.: Γ-convergence of an energy functional related to the fractional Laplacian. Calc. Var. Partial Differ. Equ. 36, 173–210 (2009)

    Article  MATH  Google Scholar 

  42. Kurzke, M.: A nonlocal singular perturbation problem with periodic well potential. ESAIM Control Optim. Calc. Var. 12(1), 52–63 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  43. Majda, A.J., Tabak, E.G.: A 2D model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Physica D 98(2–4), 515–522 (1996). Nonlinear phenomena in ocean dynamics, Los Alamos, NM (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Maz’ya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2), 230–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Preprint. http://arxiv.org/abs/0809.2455

  46. Modica, L.: Convergence to minimal surfaces problem and global solutions of Δu=2(u 3−u). In: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, pp. 223–244. Pitagora, Bologna (1979)

    Google Scholar 

  47. Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 23–142 (1987)

    Article  MathSciNet  Google Scholar 

  48. Modica, L., Mortola, S.: Un esempio di Γ −-convergenza. Boll. Unione Mat. Ital, B 14, 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  49. Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. (2012). doi:10.1007/s10231-011-0243-9

    Google Scholar 

  50. Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. 169(1), 41–78 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Savin, O., Valdinoci, E.: Elliptic PDEs with fibered nonlinearities. J. Geom. Anal. 19(2), 420–432 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Savin, O., Valdinoci, E.: Density estimates for a nonlocal variational model via the Sobolev inequality. SIAM J. Math. Anal. 43, 2675–2687 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  53. Savin, O., Valdinoci, E.: Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré, Anal. Non Linéaire (2012). doi:10.1016/j.anihpc.2012.01.006

    Google Scholar 

  54. Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. Preprint. http://www.math.columbia.edu/~savin/s-density-estimates.pdf

  55. Savin, O., Valdinoci, E.: Regularity of nonlocal minimal cones in dimension 2. Preprint. http://math.columbia.edu/~savin/cone_jan16.pdf

  56. Signorini, A.: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl. 18, 95–139 (1959). http://www.scienzadellecostruzioni.co.uk/Documenti/Signorini-Prima.pdf

    MathSciNet  MATH  Google Scholar 

  57. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256(6), 1842–1864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Toland, J.F.: The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145(1), 136–150 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  60. Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–34 (2009)

    MathSciNet  MATH  Google Scholar 

  61. Visintin, A.: Generalized coarea formula and fractal sets. Jpn. J. Ind. Appl. Math. 8, 175–201 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  62. Visintin, A.: Non convex functionals related to multi-phase systems. In: Proceedings of the Colloquium on Free Boundary Problems held in Irsee in June, 1987

    Google Scholar 

Download references

Acknowledgements

The second author is supported by the ERC grant ε Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities and the FIRB project A&B Analysis and Beyond.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050, Povo, Trento, Italy

    Giovanni Franzina

  2. Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milano, Italy

    Enrico Valdinoci

Authors
  1. Giovanni Franzina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enrico Valdinoci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Franzina .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, 50124, Italy

    Rolando Magnanini

  2. Graduate School of Information Sciences, Division of Mathematics, Tohoku University, Sendai, 980-8579, Japan

    Shigeru Sakaguchi

  3. , Dipartimento di Matematica ed Applicazio, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli, 80126, Italy

    Angelo Alvino

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Italia

About this chapter

Cite this chapter

Franzina, G., Valdinoci, E. (2013). Geometric Analysis of Fractional Phase Transition Interfaces. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_8

Download citation

Publish with us

Policies and ethics

Search

Navigation