A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

Hamel, François; Ros-Oton, Xavier; Sire, Yannick; Valdinoci, Enrico

doi:10.1016/j.anihpc.2016.01.001

Hamel, François ; Ros-Oton, Xavier ; Sire, Yannick ; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 469-482.

Résumé

We consider entire solutions to $L u = f (u)$ in $R^{2}$ , where $L$ is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator $L$ , we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2016.01.001

Classification : 45A05, 47G10, 47B34, 35R11
Mots-clés : Integral operators, Convolution kernels, Nonlocal equations, Stable solutions, One-dimensional symmetry, De Giorgi Conjecture

@article{AIHPC_2017__34_2_469_0,
     author = {Hamel, Fran\c{c}ois and Ros-Oton, Xavier and Sire, Yannick and Valdinoci, Enrico},
     title = {A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {469--482},
     publisher = {Elsevier},
     volume = {34},
     number = {2},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.01.001},
     mrnumber = {3610941},
     zbl = {1358.45004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.01.001/}
}

TY  - JOUR
AU  - Hamel, François
AU  - Ros-Oton, Xavier
AU  - Sire, Yannick
AU  - Valdinoci, Enrico
TI  - A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 469
EP  - 482
VL  - 34
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2016.01.001/
DO  - 10.1016/j.anihpc.2016.01.001
LA  - en
ID  - AIHPC_2017__34_2_469_0
ER  -

%0 Journal Article
%A Hamel, François
%A Ros-Oton, Xavier
%A Sire, Yannick
%A Valdinoci, Enrico
%T A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 469-482
%V 34
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2016.01.001/
%R 10.1016/j.anihpc.2016.01.001
%G en
%F AIHPC_2017__34_2_469_0

Hamel, François; Ros-Oton, Xavier; Sire, Yannick; Valdinoci, Enrico. A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 469-482. doi : 10.1016/j.anihpc.2016.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.01.001/

Bibliographie
Cité par

[1] Alberti, G.; Ambrosio, L.; Cabré, X. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., Volume 65 (2001) no. 1–3, pp. 9–33 | MR | Zbl

[2] Ambrosio, L.; Cabré, X. Entire solutions of semilinear elliptic equations in $R^{3}$ and a conjecture of De Giorgi, J. Am. Math. Soc., Volume 13 (2000) no. 4, pp. 725–739 | DOI | MR | Zbl

[3] Barlow, M.T.; Bass, R.F.; Gui, C. The Liouville property and a conjecture of De Giorgi, Commun. Pure Appl. Math., Volume 53 (2000) no. 8, pp. 1007–1038 | DOI | MR | Zbl

[4] Bates, P.W.; Fife, P.C.; Ren, X.; Wang, X. Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., Volume 138 (1997) no. 2, pp. 105–136 | DOI | MR | Zbl

[5] Berestycki, H.; Caffarelli, L.; Nirenberg, L. Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Volume 25 (1997) no. 1–2, pp. 69–94 | Numdam | MR | Zbl

[6] Berestycki, H.; Hamel, F.; Monneau, R. One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., Volume 103 (2000) no. 3, pp. 375–396 | DOI | MR | Zbl

[7] Bucur, C.; Valdinoci, E., Lecture Notes of the Unione Matematica Italiana, Volume vol. X 130, Springer (2016), pp. 20 | DOI | MR | Zbl

[8] Cabré, X.; Cinti, E. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete Contin. Dyn. Syst., Volume 28 (2010) no. 3, pp. 1179–1206 | DOI | MR | Zbl

[9] Cabré, X.; Cinti, E. Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differ. Equ., Volume 49 (2014) no. 1–2, pp. 233–269 | MR | Zbl

[10] Cabré, X.; Sire, Y. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, Hamiltonian estimates, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 1, pp. 23–53 | DOI | Numdam | MR | Zbl

[11] Cabré, X.; Sire, Y. Nonlinear equations for fractional Laplacians, II: Existence, uniqueness, and qualitative properties of solutions, Trans. Am. Math. Soc., Volume 367 (2015) no. 2, pp. 911–941 | DOI | MR | Zbl

[12] Cabré, X.; Solà-Morales, J. Layer solutions in a half-space for boundary reactions, Commun. Pure Appl. Math., Volume 58 (2005) no. 12, pp. 1678–1732 | DOI | MR | Zbl

[13] Caffarelli, L.; Silvestre, L. Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math., Volume 62 (2009), pp. 597–638 | DOI | MR | Zbl

[14] Chen, X. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ., Volume 2 (1997), pp. 125–160 | MR | Zbl

[15] E. Cinti, J. Serra, E. Valdinoci, Quantitative rigidity results for nonlocal phase transitions, preprint.

[16] Coville, J. Travelling fronts in asymmetric nonlocal reaction diffusion equation: the bistable and ignition case (preprint) | HAL

[17] Coville, J. Harnack type inequality for positive solution of some integral equation, Ann. Mat. Pura Appl. (4), Volume 191 (2012) no. 3, pp. 503–528 | DOI | MR | Zbl

[18] De Giorgi, E. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (1979) (Rome, May 8–12, 1978 669 p) | Zbl

[19] de la Llave, R.; Valdinoci, E. Symmetry for a Dirichlet–Neumann problem arising in water waves, Math. Res. Lett., Volume 16 (2009) no. 5–6, pp. 909–918 | MR | Zbl

[20] Del Pino, M.; Kowalczyk, M.; Wei, J. On De Giorgi's conjecture in dimension $N \geq 9$ , Ann. Math. (2), Volume 174 (2011) no. 3, pp. 1485–1569 | DOI | MR | Zbl

[21] Di Castro, A.; Kuusi, T.; Palatucci, G. Nonlocal Harnack inequalities, J. Funct. Anal., Volume 267 (2014), pp. 1807–1836 | DOI | MR | Zbl

[22] Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521–573 | DOI | MR | Zbl

[23] Farina, A. Symmetry for solutions of semilinear elliptic equations in $R^{N}$ and related conjectures, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX Ser., Rend. Lincei, Mat. Appl., Volume 10 (1999) no. 4, pp. 255–265 | MR | Zbl

[24] Farina, A.; Valdinoci, E. Recent Progress on Reaction–Diffusion Systems and Viscosity Solutions. Based on the International Conference on Reaction–Diffusion Systems and Viscosity Solutions, World Scientific, Hackensack, NJ (2009), pp. 74–96 (Taichung, Taiwan, January 3–6, 2007) | DOI | MR | Zbl

[25] Farina, A.; Valdinoci, E. 1D symmetry for solutions of semilinear and quasilinear elliptic equations, Trans. Am. Math. Soc., Volume 363 (2011) no. 2, pp. 579–609 | DOI | MR | Zbl

[26] Fife, P.C. Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, 1979 | DOI | MR | Zbl

[27] Fife, P.C. An integrodifferential analog of semilinear parabolic PDEs, Partial Differential Equations and Applications. Collected Papers in Honor of Carlo Pucci on the Occasion of his 70th Birthday, Lect. Notes Pure Appl. Math., vol. 177, Marcel Dekker, New York, NY, 1996, pp. 137–145 | MR | Zbl

[28] Ghoussoub, N.; Gui, C. On a conjecture of De Giorgi and some related problems, Math. Ann., Volume 311 (1998) no. 3, pp. 481–491 | DOI | MR | Zbl

[29] Hamel, F.; Valdinoci, E. A one-dimensional symmetry result for solutions of an integral equation of convolution type https://www.ma.utexas.edu/mp_arc/c/15/15-45.pdf

[30] Hutson, V.; Martinez, S.; Mischaikow, K.; Vickers, G.T. The evolution of dispersal, J. Math. Biol., Volume 47 (2003) no. 6, pp. 483–517 | DOI | MR | Zbl

[31] Maz'ya, V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342, Springer, 2011 | MR | Zbl

[32] Medlock, J.; Kot, M. Spreading disease: integro-differential equations old and new, Math. Biosci., Volume 184 (2003) no. 2, pp. 201–222 | DOI | MR | Zbl

[33] Palatucci, G.; Savin, O.; Valdinoci, E. Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4), Volume 192 (2013) no. 4, pp. 673–718 | DOI | MR | Zbl

[34] Ros-Oton, X.; Sire, Y. Entire solutions to semilinear nonlocal equations in $R^{2}$ , 2015 | arXiv

[35] Savin, O. Regularity of flat level sets in phase transitions, Ann. Math. (2), Volume 169 (2009) no. 1, pp. 41–78 | DOI | MR | Zbl

[36] Sire, Y.; Valdinoci, E. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., Volume 256 (2009) no. 6, pp. 1842–1864 | DOI | MR | Zbl

[37] Tan, J.; Xiong, J. A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., Volume 3 (2011) no. 3, pp. 975–983 | MR | Zbl

Cité par Sources :