Bernstein and De Giorgi type problems: new results via a geometric approach

Farina, Alberto; Sciunzi, Berardino; Valdinoci, Enrico

Farina, Alberto ; Sciunzi, Berardino ; Valdinoci, Enrico

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 741-791.

Résumé

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 (| \nabla u (x) |) \nabla 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

ℝ^{2}

and

ℝ^{3}

and of the Bernstein problem on the flatness of minimal area graphs in

ℝ^{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.

MR 2483642 | Zbl 1180.35251 | 2 citations dans Numdam

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},
     pages = {741--791},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {4},
     year = {2008},
     zbl = {1180.35251},
     mrnumber = {2483642},
     language = {en},
     url = {http://archive.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, Série 5, Tome 7 (2008) no. 4, pp. 741-791. http://archive.numdam.org/item/ASNSP_2008_5_7_4_741_0/

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