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, p. 741-791

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 div a ( | u ( x ) | ) u ( x )+f(u(x))=0. Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . 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 1-Laplacian type operators.

Classification:  32H02,  30C45
@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/

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