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Some energy estimates for stable solutions to fractional Allen–Cahn equations

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In this paper we study stable solutions to the fractional equation

$$\begin{aligned} (-\Delta )^s u =f(u), \quad |u| < 1 \quad \hbox {in }\mathbb {R}^d, \end{aligned}$$

where \(0<s<1\) and \(f:[-1,1] \rightarrow \mathbb {R}\) is a \(C^{1,\alpha }\) function for \(\alpha >\max \{0, 1-2s\}\). We obtain sharp energy estimates for \(0<s<1/2\) and rough energy estimates for \(1/2 \le s <1\). These lead to a different proof from literature of the fact that when \(d=2, \, 0<s<1\), entire stable solutions to (0.1) are 1-D solutions. The scheme used in this paper is inspired by Cinti–Serra–Valdinoci [16] which deals with stable nonlocal sets, and Figalli–Serra [26] which studies stable solutions to (0.1) for the case \(s=1/2\).

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  1. 1.

    This condition can be replaced by the more general condition that the limits of u when \(x_d \rightarrow \pm \infty \) are 2D , see [18, Theorem 8.1]


  1. 1.

    Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in \(\mathbb{R}^3\) and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)

  2. 2.

    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)

  3. 3.

    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)

  4. 4.

    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Cham (2016)

  5. 5.

    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)

  6. 6.

    Cabré, X., Cinti, E.: Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. Partial Differ. Equ. 49(1–2), 233–269 (2014)

  7. 7.

    Cabré, X.: Eleonora Cinti and Joqouim Serra. Stable nonlocal phase transitions, preprint

  8. 8.

    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31(1), 23–53 (2014)

  9. 9.

    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367(2), 911–941 (2015)

  10. 10.

    Chan, H., Liu, Y., Wei, J.: A gluing construction for fractional elliptic equations. Part I: a model problem on the catenoid (2017). arXiv:1711.03215

  11. 11.

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

  12. 12.

    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

  13. 13.

    Caffarelli, L., Stinga, P.R.: Fractional elliptic equations, Caccioppoli estimates and regularity (2017). arXiv:1409.7721v3 [math.AP]

  14. 14.

    Caffarelli, L., Valdinoci, E.: Regularity properties of nonlocal minimal surfaces via limiting arguments. Adv. Math. 248, 843–871 (2013).

  15. 15.

    Cinti, E. Flatness results for nonlocal phase transitions. In: Dipierro, S. (ed.) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol. 33. Springer, Cham (2019)

  16. 16.

    Cinti, E., Serra, J., Valdinoci, E.: Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces. J. Differential Geom. 112(3), 447–504 (2019)

  17. 17.

    del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension \(N \ge 9\). Ann. of Math. (2) 174(3), 1485–1569 (2011)

  18. 18.

    Dipierro, S., Farina, A., Valdinoci, E.: A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime. Calc. Var. Partial Differ. Equ. 57(1), 15 (2018)

  19. 19.

    Dipierro, S., Serra, J., Valdinoci, E.: Improvement of flatness for nonlocal phase transitions (2016). arXiv:1611.10105

  20. 20.

    Dipierro, S., Valdinoci, E.: Long-range phase coexistence models: recent progress on the fractional Allen–Cahn equation. arXiv:1803.03850

  21. 21.

    Di Nezza, E., Palatucci, G.: Enrico Valdinoci Hitchhiker’s guide to fractional Sobolev Spaces. arXiv:1104.4345

  22. 22.

    Farina, A.: Symmetry for solutions of semilinear elliptic equations in \(\mathbb{R}^N\) and related conjectures. Ricerche Mat. 48, 129–154 (1999). (Papers in memory of Ennio De Giorgi (Italian))

  23. 23.

    Farina, A., Valdinoci, E.: 1D symmetry for solutions of semilinear and quasilinear elliptic equations. Trans. Am. Math. Soc. 363(2), 579–609 (2011)

  24. 24.

    Farina, A., Valdinoci, E.: 1D symmetry for semilinear PDEs from the limit interface of the solution. Commun. Partial Differ. Equ. 41(4), 665–682 (2016)

  25. 25.

    Figalli, A., Jerison, D.: How to recognize convexity of a set from its marginals. J. Funct. Anal. 266(3), 1685–1701 (2014)

  26. 26.

    Figalli, A., Serra, J.: On stable solutions for boundary reactions: a De Giorgi type result in dimension \(4+1\) (2017). arXiv:1705.02781

  27. 27.

    Figalli, A., Valdinoci, E.: Regularity and Bernstein-type results for nonlocal minimal surfaces. J. Reine Angew. Math. (2015).

  28. 28.

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

  29. 29.

    Modica, L., Mortola, S.: Un esempio di-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977). (Italian, with English summary)

  30. 30.

    Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. arXiv:1104.1725

  31. 31.

    Simon, L.: Schauder estimates by scaling. Calc. Var. Partial Differ. Equ. 5, 391–407 (1997)

  32. 32.

    Savin, O.: Regularity of flat level sets in phase transitions. Ann. of Math. (2) 169(1), 41–78 (2009)

  33. 33.

    Savin, O., Valdinoci, E.: Regularity of nonlocal minimal cones in dimension 2. Calc. Var. Partial Differ. Equ. 48(1–2), 33–39 (2013).

  34. 34.

    Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. J. Math. Pures Appl. (9) 101(1), 1–26 (2014). (English, with English and French summaries)

  35. 35.

    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)

  36. 36.

    Savin, O.: Rigidity of minimizers in nonlocal phase transitions (2016). arXiv:1610.09295

  37. 37.

    Savin, O.: Rigidity of minimizers in nonlocal phase transitions II (2018). arXiv:1802.01710

  38. 38.

    Savin, O., Valdinoci, E.: \(\Gamma \)-convergence for nonlocal phase transitions. Ann. Inst. Haris Poincaré Anal. Non Linéaire 29(4), 479–500 (2012)

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This research is partially supported by NSF Grants DMS-1601885 and DMS-1901914.

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Correspondence to Changfeng Gui.

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Communicated by X. Cabre.

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Gui, C., Li, Q. Some energy estimates for stable solutions to fractional Allen–Cahn equations. Calc. Var. 59, 49 (2020).

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Mathematics Subject Classification

  • Primary 35B06
  • 35J15
  • 35J20
  • 35J91
  • 53A10