We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in and and of the Bernstein problem on the flatness of minimal area graphs in . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to very degenerate operators: as an application, we prove one-dimensional symmetry for -Laplacian type operators.
@article{ASNSP_2008_5_7_4_741_0,
author = {Farina, Alberto and Sciunzi, Berardino and Valdinoci, Enrico},
title = {Bernstein and De Giorgi type problems: new results via a geometric approach},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {4},
year = {2008},
pages = {741-791},
zbl = {1180.35251},
mrnumber = {2483642},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_4_741_0}
}
Farina, Alberto; Sciunzi, Berardino; Valdinoci, Enrico. Bernstein and De Giorgi type problems: new results via a geometric approach. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 4, pp. 741-791. http://www.numdam.org/item/ASNSP_2008_5_7_4_741_0/
[1] , and , 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 1843784 | Zbl 1121.35312
[2] and , Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000) (electronic), 725-739. | MR 1775735 | Zbl 0968.35041
[3] , and , Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997-1998), 69-94. | Numdam | MR 1655510 | Zbl 1079.35513
[4] , Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551-558. | JFM 53.0670.01 | MR 1544873
[5] , and , A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), 1457-1473. | MR 1296785 | Zbl 0819.35016
[6] and , Regularity, monotonicity and symmetry of positive solutions of -Laplace equations, J. Differential Equations 206 (2004), 483-515. | MR 2096703 | Zbl 1108.35069
[7] and , 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 1942128 | Zbl 1043.49018
[8] , Convergence problems for functionals and operators, In: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978)”, Bologna, 1979, 131-188, Pitagora Editrice. | MR 533166 | Zbl 0405.49001
[9] , local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. | MR 709038 | Zbl 0539.35027
[10] and , “Measure theory and fine properties of functions”, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. | MR 1158660 | Zbl 0804.28001
[11] , “Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires”, Habilitation à diriger des recherches, Paris VI, 2002.
[12] , One-dimensional symmetry for solutions of quasilinear equations in , Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 6 (2003), 685-692. | MR 2014827 | Zbl 1115.35045
[13] , Liouville-type theorems for elliptic problems, In: “Handbook of Differential Equations: Stationary Partial Differential Equations” M. Chipot (ed.), Vol. IV Elsevier B. V., Amsterdam, 2007, 61-116. | MR 2569331 | Zbl 1191.35128
[14] , and , Bernstein and De Giorgi type problems: new results via a geometric approach, Preliminary version of this paper, available at http://cvgmt.sns.it/papers/, 2007. | MR 2483642 | Zbl 1180.35251
[15] and , D symmetry for solutions of semilinear and quasilinear elliptic equations, Preprint, 2008. | MR 2413100 | Zbl 1228.35105
[16] and , The structure of complete stable minimal surfaces in -manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211. | MR 562550 | Zbl 0439.53060
[17] and , On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481-491. | MR 1637919 | Zbl 0918.35046
[18] and , “Elliptic partial differential equations of second order”, Classics in Mathematics, Springer-Verlag, Berlin, 2001. | MR 1814364 | Zbl 1042.35002
[19] and , “Analysis”, Vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1997. | MR 1415616 | Zbl 0966.26002
[20] , A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), 679-684. | MR 803255 | Zbl 0612.35051
[21] and , Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), 219-226. | MR 500041 | Zbl 0381.35026
[22] , Regularity of flat level sets in phase transitions, to appear in Ann. of Math. 2008. | MR 2480601 | Zbl 1180.35499
[23] , “Geometria 2”, Bollati Boringhieri, Torino, 1994.
[24] , “Singular Sets and Asymptotics in Geometric Analysis” 2007.
[25] and , Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, preprint, 2008. | Zbl 1163.35019
[26] and , Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal. 141 (1998), 375-400. | MR 1620498 | Zbl 0911.49025
[27] and , A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63-85. | MR 1650327 | Zbl 0967.53006
[28] , Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. | MR 727034 | Zbl 0488.35017
[29] , Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. | MR 474389 | Zbl 0372.35030
[30] , and , Flat level set regularity of -Laplace phase transitions, Mem. Amer. Math. Soc. 182 (2006), vi-144. | MR 2228294 | Zbl 1138.35029
