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Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

1 Universitat Politècnica de Catalunya and BGSMath, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain
2 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in “narrow” sets. The existence of the solution was already proved in part I of this work.
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Keywords integro-differential semilinear equation; saddle-shaped solution; odd symmetry; Simons cone; symmetry results

Citation: Juan-Carlos Felipe-Navarro, Tomás Sanz-Perela. Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation. Mathematics in Engineering, 2021, 3(5): 1-36. doi: 10.3934/mine.2021037


  • 1. Alberti G, Bouchitté G, Seppecher P (1998) Phase transition with the line-tension effect. Arch Ration Mech Anal 144: 1-46.    
  • 2. Barrios B, Del Pezzo L, García-Melián J, et al. (2017) Monotonicity of solutions for some nonlocal elliptic problems in half-spaces. Calc Var 56: 39.    
  • 3. Barrios B, Del Pezzo L, García-Melián J, et al. (2018) Symmetry results in the half-space for a semi-linear fractional Laplace equation. Ann Mat Pur Appl 197: 1385-1416.    
  • 4. Barrios B, Peral I, Soria F, et al. (2014) A Widder's type theorem for the heat equation with nonlocal diffusion. Arch Ration Mech Anal 213: 629-650.    
  • 5. Berestycki H, Hamel F, Monneau R (2000) One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math J 103: 375-396.    
  • 6. Berestycki H, Hamel F, Nadirashvili N (2010) The speed of propagation for KPP type problems II: General domains. J Am Math Soc 23: 1-34.    
  • 7. Bucur C, Valdinoci E (2016) Nonlocal Diffusion and Applications, Springer International Publishing.
  • 8. Cabré X (1995) On the Alexandro ff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Commun Pure Appl Math 48: 539-570.    
  • 9. Cabré X (2002) Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete Contin Dyn Syst 8: 331-359.    
  • 10. Cabré X (2012) Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation. J Math Pure Appl 98: 239-256.    
  • 11. Cabré X, Sire Y (2015) Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions. T Am Math Soc 367: 911-941.
  • 12. Cabré X, Solà-Morales J (2005) Layer solutions in a half-space for boundary reactions. Commun Pure Appl Math 58: 1678-1732.    
  • 13. Cabré X, Terra J (2010) Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Commun Part Diff Eq 35: 1923-1957.    
  • 14. Chen W, Li C, Li Y (2017) A direct method of moving planes for the fractional Laplacian. Adv Math 308: 404-437.    
  • 15. Chen W, Li Y, Zhang R (2017) A direct method of moving spheres on fractional order equations. J Funct Anal 272: 4131-4157.    
  • 16. Cinti E (2013) Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian. Ann Sc Norm Super Pisa Cl Sci 12: 623-664.
  • 17. Cinti E (2018) Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete Contin Dyn Syst Ser S 11: 441-463.
  • 18. Cozzi M (2017) Regularity results and Harnack inequalities for minimizers and Solutions of nonlocal problems: A unified approach via fractional De Giorgi classes. J Funct Anal 272: 4762-4837.    
  • 19. Cozzi M (2019) Fractional De Giorgi classes and applications to nonlocal regularity theory, In: Contemporary Research in Elliptic PDEs and Related Topics, Cham: Springer, 277-299.
  • 20. Cozzi M, Figalli A (2017) Regularity theory for local and nonlocal minimal surfaces: An overview, In: Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Cham: Springer, 117-158.
  • 21. Cozzi M, Passalacqua T (2016) One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels. J Differ Equations 260: 6638-6696.    
  • 22. Dávila J, del Pino M, Wei J (2018) Nonlocal s-minimal surfaces and Lawson cones. J Differ Geom 109: 111-175.    
  • 23. del Pino M, Kowalczyk M, Wei J (2011) On De Giorgi's conjecture in dimension N ≥ 9. Ann Math 174: 1485-1569.    
  • 24. Dipierro S, Soave N, Valdinoci E (2017) On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math Ann 369: 1283-1326.    
  • 25. Evans LC (2010) Partial Differential Equations, 2 Eds., American Mathematical Society.
  • 26. Fall M, Weth T (2016) Monotonicity and nonexistence results for some fractional elliptic problems in the half-space. Commun Contemp Math 18: 1550012.    
  • 27. Farina A, Valdinoci E (2009) The state of the art for a conjecture of De Giorgi and related problems, In: Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, Hackensack: World Sci. Publ., 74-96.
  • 28. Felipe-Navarro JC, Sanz-Perela T (2018) Uniqueness and stability of the saddle-shaped solution to the fractional Allen-Cahn equation. Rev Mat Iberoam DOI: 10.4171/rmi/1185.
  • 29. Felipe-Navarro JC, Sanz-Perela T (2020) Semilinear integro-differential equations, I: odd solutions with respect to the Simons cone. J Funct Anal 278: 108309.    
  • 30. Felmer P, Wang Y (2014) Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun Contemp Math 16: 1350023.    
  • 31. González MdM (2009) Gamma convergence of an energy functional related to the fractional Laplacian. Calc Var 36: 173-210.    
  • 32. Hamel F, Ros-Oton X, Sire Y, et al. (2017) A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Ann I H Poincaré Non Linear Anal 34: 469-482.
  • 33. Jerison D, Monneau R (2004) Towards a counter-example to a conjecture of De Giorgi in high dimensions. Ann Mat Pur Appl 183: 439-467.    
  • 34. Li Y, Zhang L (2003) Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations. J Anal Math 90: 27-87.    
  • 35. Liu Y, Wang K, Wei J (2020) Stability of the saddle solutions for the Allen-Cahn equation. arXiv 2001.07356.
  • 36. Quaas A, Xia A (2015) Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space. Calc Var 52: 641-659.    
  • 37. Ros-Oton X (2016) Nonlocal elliptic equations in bounded domains: A survey. Publ Mat 60: 3-26.    
  • 38. Ros-Oton X, Serra J (2016) Regularity theory for general stable operators. J Differ Equations 260: 8675-8715.    
  • 39. Sanz-Perela T (2019) Stable solutions of nonlinear fractional elliptic problems, Ph.D. thesis of Universitat Politècnica de Catalunya.
  • 40. Savin O, Valdinoci E (2012) Γ-convergence for nonlocal phase transitions. Ann I H Poincaré Non Linéaire Anal 29: 479-500.
  • 41. Savin O, Valdinoci E (2013) Regularity of nonlocal minimal cones in dimension 2. Calc Var 48: 33-39.    
  • 42. Serra J (2015) Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels. Calc Var 54: 3571-3601.    
  • 43. Servadei R, Valdinoci E (2013) Variational methods for non-local operators of elliptic type. Discrete Contin Dyn Syst 33: 2105-2137.    
  • 44. Valdinoci E (2013) A fractional framework for perimeters and phase transitions. Milan J Math 81: 1-23.    


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