We study nonlinear elliptic equations of the form where the main assumption on and is that there exists a one dimensional solution which solves the equation in all the directions . We show that entire monotone solutions are one dimensional if their level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.
@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}, pages = {369--405}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {3}, year = {2008}, mrnumber = {2466434}, zbl = {1181.35111}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_3_369_0/} }
TY - JOUR AU - Savin, Ovidiu TI - Entire solutions to a class of fully nonlinear elliptic equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 SP - 369 EP - 405 VL - 7 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_3_369_0/ LA - en ID - ASNSP_2008_5_7_3_369_0 ER -
%0 Journal Article %A Savin, Ovidiu %T Entire solutions to a class of fully nonlinear elliptic equations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2008 %P 369-405 %V 7 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2008_5_7_3_369_0/ %G en %F 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, Serie 5, Volume 7 (2008) no. 3, pp. 369-405. http://archive.numdam.org/item/ASNSP_2008_5_7_3_369_0/
[1] Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), 725-739. | MR | Zbl
and ,[2] The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math. 53 (2000), 1007-1038. | MR | Zbl
, and ,[3] R., One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000), 375-396. | MR | Zbl
, and[4] “Fully Nonlinear Elliptic Equations”, American Mathematical Society, Colloquium Publications 43, Providence, RI, 1995. | MR | Zbl
and ,[5] Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995), 1-12. | MR | Zbl
and ,[6] Phase transitions: uniform regularity of the intermediate layers, J. Reine Angew. Math. 593 (2006), 209-235. | MR | Zbl
and ,[7] A Harnack inequality approach to the interior regularity of elliptic equations, Indiana Univ. Math. J. 42 (1993), 145-157. | MR | Zbl
and ,[8] Convergence problems for functional and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (1978), 131-188. | MR | Zbl
,[9] Symmetry of global solutions to fully nonlinear equations in 2D, Indiana Univ. Math. J., to appear. | MR | Zbl
and ,[10] C., On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481-491. | MR | Zbl
and[11] -convergence to minimal surfaces problem and global solutions of , Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, 223-244, Pitagora, Bologna, 1979. | MR | Zbl
,[12] Regularity of flat level sets for phase transitions, Ann. of Math., to appear. | Zbl
,[13] Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007) 557-578. | MR | Zbl
,[14] “Flat Level Set Regularity of -Laplace Phase Transitions”, Mem. Amer. Math. Soc., Vol. 182, 2006. | MR | Zbl
, and ,