# American Institute of Mathematical Sciences

June  2018, 11(3): 441-463. doi: 10.3934/dcdss.2018024

## Saddle-shaped solutions for the fractional Allen-Cahn equation

 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5,40126 Bologna, Italy

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: The author is supported by MINECO grant MTM2014-52402-C3-1-P, the ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Interacting Dynamical Systems, the GNAMPA project Metodi variazionali per problemi nonlocali and is part of the Catalan research group 2014 SGR 1083.

We establish existence and qualitative properties of solutions to the fractional Allen-Cahn equation, which vanish on the Simons cone and are even with respect to the coordinate axes. These solutions are called saddle-shaped solutions.

More precisely, we prove monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4, 6$. We extend to any fractional power $s$ of the Laplacian, some results obtained for the case $s=1/2$ in [19].

The interest in the study of saddle-shaped solutions comes in connection with a celebrated De Giorgi conjecture on the one-dimensional symmetry of monotone solutions and of minimizers for the Allen-Cahn equation. Saddle-shaped solutions are candidates to be (not one-dimensional) minimizers in high dimension, a property which is not known to hold yet.

Citation: Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
##### References:
 [1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33.  doi: 10.1023/A:1010602715526. [2] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb R^3$ and a Conjecture of De Giorgi, Journal Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3. [3] V. Banica, M. D. M. Gonzalez and M. Saez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam., 31 (2015), 681-712.  doi: 10.4171/RMI/850. [4] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309. [5] C. Brändle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [6] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications Lecture Notes of the Unione Matematica Italiana, 20 Springer, Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [7] X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation, Journal de Mathématiques Pures et Appliquées, 98 (2012), 239-256.  doi: 10.1016/j.matpur.2012.02.006. [8] X. Cabré and E. Cinti, Energy estimates and 1D symmetry for nonlinear equations involving the half-Laplacian, Discrete and Continuous Dynamical Systems, 28 (2010), 1179-1206.  doi: 10.3934/dcds.2010.28.1179. [9] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. of Var. and PDE, 49 (2014), 233-269.  doi: 10.1007/s00526-012-0580-6. [10] X. Cabré, E. Cinti and J. Serra, Stable $s$-minimal cones in $\mathbb R^3$ are flat for $s~ 1$, available at https://arxiv.org/abs/1710.08722. [11] X. Cabré, E. Cinti and J. Serra, Stable nonlocal phase transitions, forthcoming. [12] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [13] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0. [14] X. Cabré and J. Solá-Morales, Layer Solutions in a Halph-Space for Boundary reactions, Comm. Pure and Appl. Math., 58 (2005), 1678-1732.  doi: 10.1002/cpa.20093. [15] X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb R^{2m}$, J. Eur. Math. Soc., 11 (2009), 819-843.  doi: 10.4171/JEMS/168. [16] X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. in Partial Differential Equations, 35 (2010), 1923-1957.  doi: 10.1080/03605302.2010.484039. [17] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [18] L. Caffarelli and L. Silvestre, An extension related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [19] E. Cinti, Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 623-664. [20] E. Cinti, J. Davila and M. Del Pino, Solutions of the fractional Allen-Cahn equation which are invariant under screw motion, J. Lond. Math. Soc., 94 (2016), 295-313.  doi: 10.1112/jlms/jdw033. [21] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces, to appear in J. Diff. Geom. [22] H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew Math. Phys., 43 (1992), 984-998.  doi: 10.1007/BF00916424. [23] J. Davila, M. Del Pino and J. Wei, Nonlocal $s$-minimal surfaces and Lawson cones, to appear in J. Diff. Geom. [24] M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi Conjecture in dimension $N≥q 9$, Ann. of Math., 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3. [25] S. Dipierro, J. Serra and E. Valdinoci, Improvement of flatness for nonlocal phase transitions, preprint, arXiv: 1611.10105. [26] S. Dipierro, A. Farina and E. Valdinoci, A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime, preprint, arXiv: 1705.00320. [27] A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., 729 (2017), 263-273.  doi: 10.1515/crelle-2015-0006. [28] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196. [29] Y. Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [30] Y. Liu, K. Wang and J. Wei, Global minimizers of the Allen-Cahn equation in dimension $n≥q 8$, to appear in Journal de Mathématiques Pures et Appliquées. [31] O. Savin, Phase ransitions: Regularity of flat level sets, Ann. of Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41. [32] O. Savin, Rigidity of minimizers in nonlocal phase transitions, preprint, arXiv: 1610.09295. [33] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. of Var. and PDE, 48 (2013), 33-39.  doi: 10.1007/s00526-012-0539-7. [34] O. Savin and E. Valdinoci, $Γ$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006. [35] M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.  doi: 10.1017/S0308210500030493. [36] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, Jour. Functional Analysis, 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020. [37] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859.  doi: 10.3934/dcds.2013.33.837.

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##### References:
 [1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33.  doi: 10.1023/A:1010602715526. [2] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb R^3$ and a Conjecture of De Giorgi, Journal Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3. [3] V. Banica, M. D. M. Gonzalez and M. Saez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam., 31 (2015), 681-712.  doi: 10.4171/RMI/850. [4] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309. [5] C. Brändle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [6] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications Lecture Notes of the Unione Matematica Italiana, 20 Springer, Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [7] X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation, Journal de Mathématiques Pures et Appliquées, 98 (2012), 239-256.  doi: 10.1016/j.matpur.2012.02.006. [8] X. Cabré and E. Cinti, Energy estimates and 1D symmetry for nonlinear equations involving the half-Laplacian, Discrete and Continuous Dynamical Systems, 28 (2010), 1179-1206.  doi: 10.3934/dcds.2010.28.1179. [9] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. of Var. and PDE, 49 (2014), 233-269.  doi: 10.1007/s00526-012-0580-6. [10] X. Cabré, E. Cinti and J. Serra, Stable $s$-minimal cones in $\mathbb R^3$ are flat for $s~ 1$, available at https://arxiv.org/abs/1710.08722. [11] X. Cabré, E. Cinti and J. Serra, Stable nonlocal phase transitions, forthcoming. [12] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [13] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0. [14] X. Cabré and J. Solá-Morales, Layer Solutions in a Halph-Space for Boundary reactions, Comm. Pure and Appl. Math., 58 (2005), 1678-1732.  doi: 10.1002/cpa.20093. [15] X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb R^{2m}$, J. Eur. Math. Soc., 11 (2009), 819-843.  doi: 10.4171/JEMS/168. [16] X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. in Partial Differential Equations, 35 (2010), 1923-1957.  doi: 10.1080/03605302.2010.484039. [17] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [18] L. Caffarelli and L. Silvestre, An extension related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [19] E. Cinti, Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 623-664. [20] E. Cinti, J. Davila and M. Del Pino, Solutions of the fractional Allen-Cahn equation which are invariant under screw motion, J. Lond. Math. Soc., 94 (2016), 295-313.  doi: 10.1112/jlms/jdw033. [21] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces, to appear in J. Diff. Geom. [22] H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew Math. Phys., 43 (1992), 984-998.  doi: 10.1007/BF00916424. [23] J. Davila, M. Del Pino and J. Wei, Nonlocal $s$-minimal surfaces and Lawson cones, to appear in J. Diff. Geom. [24] M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi Conjecture in dimension $N≥q 9$, Ann. of Math., 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3. [25] S. Dipierro, J. Serra and E. Valdinoci, Improvement of flatness for nonlocal phase transitions, preprint, arXiv: 1611.10105. [26] S. Dipierro, A. Farina and E. Valdinoci, A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime, preprint, arXiv: 1705.00320. [27] A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., 729 (2017), 263-273.  doi: 10.1515/crelle-2015-0006. [28] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196. [29] Y. Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [30] Y. Liu, K. Wang and J. Wei, Global minimizers of the Allen-Cahn equation in dimension $n≥q 8$, to appear in Journal de Mathématiques Pures et Appliquées. [31] O. Savin, Phase ransitions: Regularity of flat level sets, Ann. of Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41. [32] O. Savin, Rigidity of minimizers in nonlocal phase transitions, preprint, arXiv: 1610.09295. [33] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. of Var. and PDE, 48 (2013), 33-39.  doi: 10.1007/s00526-012-0539-7. [34] O. Savin and E. Valdinoci, $Γ$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006. [35] M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1241-1275.  doi: 10.1017/S0308210500030493. [36] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, Jour. Functional Analysis, 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020. [37] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859.  doi: 10.3934/dcds.2013.33.837.
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