# 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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [11] X. Cabré, E. Cinti and J. Serra, Stable nonlocal phase transitions, forthcoming.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [23] J. Davila, M. Del Pino and J. Wei, Nonlocal $s$-minimal surfaces and Lawson cones, to appear in J. Diff. Geom. Google Scholar [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. Google Scholar [25] S. Dipierro, J. Serra and E. Valdinoci, Improvement of flatness for nonlocal phase transitions, preprint, arXiv: 1611.10105.Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [31] O. Savin, Phase ransitions: Regularity of flat level sets, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41. Google Scholar [32] O. Savin, Rigidity of minimizers in nonlocal phase transitions, preprint, arXiv: 1610.09295.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [11] X. Cabré, E. Cinti and J. Serra, Stable nonlocal phase transitions, forthcoming.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [23] J. Davila, M. Del Pino and J. Wei, Nonlocal $s$-minimal surfaces and Lawson cones, to appear in J. Diff. Geom. Google Scholar [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. Google Scholar [25] S. Dipierro, J. Serra and E. Valdinoci, Improvement of flatness for nonlocal phase transitions, preprint, arXiv: 1611.10105.Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [31] O. Savin, Phase ransitions: Regularity of flat level sets, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41. Google Scholar [32] O. Savin, Rigidity of minimizers in nonlocal phase transitions, preprint, arXiv: 1610.09295.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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