Entire solutions to a class of fully nonlinear elliptic equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, p. 369-405
We study nonlinear elliptic equations of the form F(D 2 u)=f(u) where the main assumption on F and f is that there exists a one dimensional solution which solves the equation in all the directions ξ n . We show that entire monotone solutions u are one dimensional if their 0 level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.
Classification:  35J70,  35B65
@article{ASNSP_2008_5_7_3_369_0,
     author = {Savin, Ovidiu},
     title = {Entire solutions to a class of fully nonlinear elliptic equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {3},
     year = {2008},
     pages = {369-405},
     zbl = {1181.35111},
     mrnumber = {2466434},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2008_5_7_3_369_0}
}
Savin, Ovidiu. Entire solutions to a class of fully nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, pp. 369-405. http://www.numdam.org/item/ASNSP_2008_5_7_3_369_0/

[1] L. Ambrosio and X. Cabrè, Entire solutions of semilinear elliptic equations in 3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), 725-739. | MR 1775735 | Zbl 0968.35041

[2] M. Barlow, R. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math. 53 (2000), 1007-1038. | MR 1755949 | Zbl 1072.35526

[3] H. Berestycki, F. Hamel and R. Monneau R., One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000), 375-396. | MR 1763653 | Zbl 0954.35056

[4] L. Caffarelli and X. Cabrè, “Fully Nonlinear Elliptic Equations”, American Mathematical Society, Colloquium Publications 43, Providence, RI, 1995. | MR 1351007 | Zbl 0834.35002

[5] L. Caffarelli and A. Cordoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995), 1-12. | MR 1310848 | Zbl 0829.49013

[6] L. Caffarelli and A. Cordoba, Phase transitions: uniform regularity of the intermediate layers, J. Reine Angew. Math. 593 (2006), 209-235. | MR 2227143 | Zbl 1090.49019

[7] L. Caffarelli and L. Wang, A Harnack inequality approach to the interior regularity of elliptic equations, Indiana Univ. Math. J. 42 (1993), 145-157. | MR 1218709 | Zbl 0810.35023

[8] E. De Giorgi, Convergence problems for functional and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (1978), 131-188. | MR 533166 | Zbl 0405.49001

[9] D. De Silva and O. Savin, Symmetry of global solutions to fully nonlinear equations in 2D, Indiana Univ. Math. J., to appear. | MR 2504413 | Zbl 1165.35021

[10] N. Ghoussoub and C. Gui C., On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481-491. | MR 1637919 | Zbl 0918.35046

[11] L. Modica, Γ-convergence to minimal surfaces problem and global solutions of Δu=2(u 3 -u), Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, 223-244, Pitagora, Bologna, 1979. | MR 533169 | Zbl 0408.49041

[12] O. Savin, Regularity of flat level sets for phase transitions, Ann. of Math., to appear. | Zbl 1180.35499

[13] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007) 557-578. | MR 2334822 | Zbl 1221.35154

[14] E. Valdinoci, B. Sciunzi and O. Savin, “Flat Level Set Regularity of p-Laplace Phase Transitions”, Mem. Amer. Math. Soc., Vol. 182, 2006. | MR 2228294 | Zbl 1138.35029