We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.
Mots-clés : Nonlocal operators, Nonlinear operators, Stable solutions, Classifications of solutions, One-dimensional solutions
@article{AIHPC_2019__36_2_523_0, author = {Fazly, Mostafa and Sire, Yannick}, title = {Symmetry properties for solutions of nonlocal equations involving nonlinear operators}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {523--543}, publisher = {Elsevier}, volume = {36}, number = {2}, year = {2019}, doi = {10.1016/j.anihpc.2018.07.004}, mrnumber = {3913197}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.07.004/} }
TY - JOUR AU - Fazly, Mostafa AU - Sire, Yannick TI - Symmetry properties for solutions of nonlocal equations involving nonlinear operators JO - Annales de l'I.H.P. Analyse non linéaire PY - 2019 SP - 523 EP - 543 VL - 36 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2018.07.004/ DO - 10.1016/j.anihpc.2018.07.004 LA - en ID - AIHPC_2019__36_2_523_0 ER -
%0 Journal Article %A Fazly, Mostafa %A Sire, Yannick %T Symmetry properties for solutions of nonlocal equations involving nonlinear operators %J Annales de l'I.H.P. Analyse non linéaire %D 2019 %P 523-543 %V 36 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2018.07.004/ %R 10.1016/j.anihpc.2018.07.004 %G en %F AIHPC_2019__36_2_523_0
Fazly, Mostafa; Sire, Yannick. Symmetry properties for solutions of nonlocal equations involving nonlinear operators. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 523-543. doi : 10.1016/j.anihpc.2018.07.004. http://www.numdam.org/articles/10.1016/j.anihpc.2018.07.004/
[1] 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), pp. 9–33 | DOI | MR | Zbl
[2] Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi, J. Am. Math. Soc., Volume 13 (2000), pp. 725–739 | DOI | MR | Zbl
[3] On the Liouville property for divergence form operators, Can. J. Math., Volume 50 (1998), pp. 487–496 | DOI | MR | Zbl
[4] The Liouville property and a conjecture of De Giorgi, Commun. Pure Appl. Math., Volume 53 (2000), pp. 1007–1038 | DOI | MR | Zbl
[5] Harnack inequalities for jump processes, Potential Anal., Volume 17 (2002), pp. 375–388 | MR | Zbl
[6] Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 25 (1997), pp. 69–94 | Numdam | MR | Zbl
[7] Non-local gradient dependent operators, Adv. Math., Volume 230 (2012), pp. 1859–1894 | DOI | MR | Zbl
[8] A symmetry result in for global minimizers of a general type of nonlocal energy | arXiv | DOI
[9] Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete Contin. Dyn. Syst., Volume 28 (2010), pp. 1179–1206 | DOI | MR | Zbl
[10] Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differ. Equ., Volume 49 (2014), pp. 233–269 | DOI | MR | Zbl
[11] Layer solutions in a half-space for boundary reactions, Commun. Pure Appl. Math., Volume 58 (2005), pp. 1678–1732 | DOI | MR | Zbl
[12] An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions, Nonlinear Anal., Volume 137 (2016), pp. 246–265 | DOI | MR
[13] Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Trans. Am. Math. Soc., Volume 367 (2015), pp. 911–941 | MR | Zbl
[14] An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 1245–1260 | DOI | MR | Zbl
[15] Local behavior of fractional p-minimizers, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016), pp. 1279–1299 | DOI | Numdam | MR
[16] Nonlocal Harnack inequalities, J. Funct. Anal., Volume 267 (2014), pp. 1807–1836 | DOI | MR | Zbl
[17] Geometric inequalities for fractional Laplace operators and applications, NoDEA Nonlinear Differ. Equ. Appl., Volume 22 (2015), pp. 1699–1714 | DOI | MR
[18] One-dimensional solutions of non-local Allen–Cahn-type equations with rough kernels, J. Differ. Equ., Volume 260 (2016), pp. 6638–6696 | DOI | MR
[19] Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Pitagora, Bologna (1979), pp. 131–188 (Rome, 1978) | MR | Zbl
[20] On De Giorgi's conjecture in dimension , Ann. Math. (2), Volume 174 (2011), pp. 1485–1569 | DOI | MR | Zbl
[21] Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 7 (2008), pp. 741–791 | Numdam | MR | Zbl
[22] On a Poincaré type formula for solutions of singular and degenerate elliptic equations, Manuscr. Math., Volume 132 (2010), pp. 335–342 | DOI | MR | Zbl
[23] M. Fazly, C. Gui, On nonlocal systems with jump processes of finite range and with decays, Preprint, 2018. | MR
[24] On a conjecture of De Giorgi and some related problems, Math. Ann., Volume 311 (1998) no. 3, pp. 481–491 | DOI | MR | Zbl
[25] About De Giorgi's conjecture in dimensions 4 and 5, Ann. Math. (2), Volume 157 (2003), pp. 313–334 | DOI | MR | Zbl
[26] A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 2, pp. 469–482 | Numdam | MR | Zbl
[27] Nonlocal equations with measure data, Commun. Math. Phys., Volume 337 (2015) no. 3, pp. 1317–1368 | DOI | MR
[28] Regularity of flat level sets in phase transitions, Ann. Math. (2), Volume 169 (2009) no. 1, pp. 41–78 | DOI | MR | Zbl
[29] Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., Volume 55 (2006), pp. 1155–1174 | DOI | MR | Zbl
[30] Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., Volume 256 (2009), pp. 1842–1864 | DOI | MR | Zbl
[31] A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., Volume 503 (1998), pp. 63–85 | MR | Zbl
[32] Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal., Volume 141 (1998), pp. 375–400 | DOI | MR | Zbl
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