On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 181-197.

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in 2 . We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

Classification : 35J40, 35J30, 35J65, 35R35, 35J99
@article{ASNSP_2003_5_2_1_181_0,
     author = {Dolbeault, Jean and Monneau, R\'egis},
     title = {On a {Liouville} type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {181--197},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     mrnumber = {1990978},
     zbl = {1170.35380},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_181_0/}
}
TY  - JOUR
AU  - Dolbeault, Jean
AU  - Monneau, Régis
TI  - On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2003
SP  - 181
EP  - 197
VL  - 2
IS  - 1
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2003_5_2_1_181_0/
LA  - en
ID  - ASNSP_2003_5_2_1_181_0
ER  - 
%0 Journal Article
%A Dolbeault, Jean
%A Monneau, Régis
%T On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2003
%P 181-197
%V 2
%N 1
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2003_5_2_1_181_0/
%G en
%F ASNSP_2003_5_2_1_181_0
Dolbeault, Jean; Monneau, Régis. On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 181-197. http://www.numdam.org/item/ASNSP_2003_5_2_1_181_0/

[1] S. Alama - L. Bronsard - C. Gui, Stationary layered solutions in n for an Allen-Cahn system with multiple well potential, Calc. Var. Partial Differential Equations 5 (1997), 359-390. | MR | Zbl

[2] G. Alberti - L. Ambrosio - 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

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

[4] S.B. Angenent, Uniqueness of the solution of a semilinear boundary value problem, Math. Ann. 272 (1985), 129-138. | MR | Zbl

[5] M.T. Barlow, On the Liouville property for divergence form operators, Canad. J. Math. 50 (1998), 487-496. | MR | Zbl

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

[7] H. Berestycki - L. Caffarelli - L. Nirenberg, Symmetry for elliptic equations in a half space, in “Boundary value problems for partial differential equations and applications", J.-L. Lions et al. (eds.), RMA Res. Notes Appl. Math., Masson, Paris, 1993, pp. 27-42. | MR | Zbl

[8] H. Berestycki - L. Caffarelli - L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 (1997), 1089-1111. | MR | Zbl

[9] H. Berestycki - L. Caffarelli - L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, (4) (1997), 69-94. | Numdam | MR | Zbl

[10] H. Berestycki - L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. 22 (1991), 1-37. | MR | Zbl

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

[12] S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom ellipschen Typus, Math. Zeit. 26, 551-558 (1927), translation of the original version: Comm. de la Soc. Math. de Kharkov 2ème sér. [Zap. Harkov. Mat. Obsc. (2)] 15 (1915-1917), 38-45. | JFM | MR

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

[14] X. Cabré, A conjecture of De Giorgi on symmetry for elliptic equations in n , European Congress of Mathematics, Vol. I (Barcelona, 2000), 259-265, Progr. Math., 201, Birkhäuser, Basel, (2001). | MR | Zbl

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

[16] L. A. Caffarelli - A. Cordoba, Phase transition: uniform regularity of the transition layers, preprint (2001).

[17] L. Caffarelli - N. Garofalo - F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), 1457-1473. | MR | Zbl

[18] D. Danielli - N. Garofalo, Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions, Calc. Var. Partial Differential Equations 15 (2002), 451-491. | MR | Zbl

[19] E. De Giorgi, Convergence problems for functionals and operators, in “Proc. Int. Meeting on Recent Methods in Nonlinear Analysis", Rome, 1978, E. De Giorgi et al. (eds.), Pitagora, Bologna, 1979, pp. 131-188. | MR | Zbl

[20] J. Dolbeault - R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems [Estimations de convexité pour des équations elliptiques non-linéaires et application à des problèmes de frontière libre], C. R. Acad. Sci., Paris, Sér. I. Math. 331 (2000), 771-776. | MR | Zbl

[21] J. Dolbeault - R. Monneau, Convexity estimates for nonlinear elliptic equations and application to free boundary problems, Ann. Inst. Henri Poincaré Anal. Non Linéaire 19 (2002), 903-926. | Numdam | MR | Zbl

[22] A. Farina, Some remarks on a conjecture of De Giorgi, Calc. Var. Partial Differential Equations 8 (3) (1999), 233-245. | MR | Zbl

[23] A. Farina, Symmetry for solutions of semilinear elliptic equations in n and related conjectures, Ricerche Mat. 48 (1999), 129-154. | MR | Zbl

[24] A. Farina, One-dimensional symmetry for solutions of quasilinear equations in 2 , to appear in Bol. Un. Mat. Ital. (2002). | MR | Zbl

[25] A. Farina, Propriétés de monotonie et de symétrie unidimensionnelle pour les solutions de Δu+f(u)=0 avec des fonctions f éventuellement discontinues [Monotonicity and one-dimensional symmetry for the solutions of Δu+f(u)=0 with possibly discontinuous nonlinearity], C. R. Acad. Sci., Paris, Sér. I. Math. 330 (2000), 973-978. | MR | Zbl

[26] A. Farina, Monotonicity and one-dimensional symmetry for the solutions of Δu+f(u)=0 in N with possibly discontinuous nonlinearity, Adv. Math. Sci. Appl. 11 (2001), 811-834. | MR | Zbl

[27] A. Farina, Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of R N and in half spaces, to appear in Adv. Math. Sci. Appl. 13, (2003). | MR | Zbl

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

[29] N. Ghoussoub - C. Gui, On De Giorgi's Conjecture in Dimensions 4 and 5, to appear in Ann. of Math. | MR

[30] E. Giusti, “Minimal surfaces and functions of bounded variation”, Birkhäuser Verlag, Basel, Boston, 1984. | MR | Zbl

[31] D. Jerison - R. Monneau, The existence of a symmetric global minimizer on n-1 implies the existence of a counter-example to a conjecture of De Giorgi in n [L'existence d'un minimiseur global symétrique sur n-1 implique l'existence d'un contre-exemple à une conjecture de De Giorgi dans n ], C. R. Acad. Sci., Paris, Sér. I Math. 333 (2001), 427-431. | MR | Zbl

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

[33] L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, “Partial differential equations and the calculus of variations", Vol. II, Progr. Nonlinear Differential Equations Appl. 2, Birkhäuser, Boston, 1989, pp. 843-850. | MR | Zbl

[34] L. Modica - S. Mortola, Some entire solutions in the plane of nonlinear Poisson equations, Boll. Un. Mat. Ital. 5 (1980), 614-622. | MR | Zbl

[35] C.B. Morrey, “Multiple integrals in the caculus of variations”, Springer-Verlag, Berlin, Heidelberg, New-York, 1966. | MR | Zbl

[36] L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math. 6 (1953), 103-156. | MR | Zbl

[37] J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971), 304-318. | MR | Zbl

[38] L. Simon, Entire solutions of the minimal surface equation, J. Differential Geom. 30 (1989), 643-688. | MR | Zbl

[39] L. Simon, The minimal surface equation, “Geometry", V, 239-272, Encyclopaedia Math. Sci. 90, Springer, Berlin, 1997. | MR | Zbl

[40] R.P. Sperb, “Maximum principles and their applications”, Mathematics in Science and Engineering, 157. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New-York, London, 1981. | MR | Zbl