Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 623-664.

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (-Δ) 1/2 u=f(u) in the whole space 2m , where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate.

More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions 2m=4 and 2m=6.

These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.

Publié le :
Classification : 35J61, 35J20, 35B40, 35B08
@article{ASNSP_2013_5_12_3_623_0,
     author = {Cinti, Eleonora},
     title = {Saddle-shaped solutions of bistable elliptic equations involving the {half-Laplacian}},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {623--664},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {3},
     year = {2013},
     mrnumber = {3137458},
     zbl = {1283.35042},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2013_5_12_3_623_0/}
}
TY  - JOUR
AU  - Cinti, Eleonora
TI  - Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2013
SP  - 623
EP  - 664
VL  - 12
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2013_5_12_3_623_0/
LA  - en
ID  - ASNSP_2013_5_12_3_623_0
ER  - 
%0 Journal Article
%A Cinti, Eleonora
%T Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2013
%P 623-664
%V 12
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2013_5_12_3_623_0/
%G en
%F ASNSP_2013_5_12_3_623_0
Cinti, Eleonora. Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 623-664. http://archive.numdam.org/item/ASNSP_2013_5_12_3_623_0/

[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. | MR | Zbl

[2] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in 3 and a Conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), 725–739. | MR | Zbl

[3] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. | EuDML | MR | Zbl

[4] X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete Contin. Dyn. Syst. 28 (2010), 1179–1206. A special issue dedicated to Louis Nirenberg on the Occasion of his 85th Birthday. | MR | Zbl

[5] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations, to appear (DOI: 10.1007/s00526-012-0580-6). | MR | Zbl

[6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates, accepted for publication in Trans. Amer. Mat. Soc. | Numdam | MR | Zbl

[7] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, forthcoming. | MR

[8] X. Cabré and J. Solà-Morales, Layer solutions in a halph-space for boundary reactions, Comm. Pure and Appl. Math. 58 (2005), 1678–1732. | MR | Zbl

[9] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093. | MR | Zbl

[10] X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of 2m , J. Eur. Math. Soc. 11 (2009), 819–843. | EuDML | MR | Zbl

[11] X. Cabré and J. Terra, Qualitative properties of saddle-shaped solutions to bistable diffusion equations, Comm. Partial Differential Equations 35 (2010), 1923–1957. | MR | Zbl

[12] L. Caffarelli and L. Silvestre, An estension related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260. | MR | Zbl

[13] H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew Math. Phys. 43 (1992), 984–998. | MR | Zbl

[14] M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension N9, Ann. of Math. 174 (2011), 1485–1569. | MR | Zbl

[15] J. P. Garcia Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476. | MR | Zbl

[16] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481–491. | MR | Zbl

[17] D. Jerison and R. Monneau, Towards a counter-example to a conjecture of De Giorgi in high dimensions, Ann. Mat. Pura Appl. 183 (2004), 439–467. | MR | Zbl

[18] Y. Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87. | MR | Zbl

[19] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), 679–684. | MR | Zbl

[20] O. Savin, Phase transitions: regularity of flat level sets, Ann. of Math. 169 (2009), 41–78. | MR | Zbl

[21] M. Schatzman, On the stability of the saddle solution of Allen-Cahn’s equation, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1241–1275. | MR | Zbl

[22] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 1842–1864. | MR | Zbl

[23] J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal. 145 (1997), 136–150. | MR | Zbl