We use a Poincare 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(vertical bar del u(x)vertical bar)del 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.
Bernstein and De Giorgi type problems: new results via a geometric approach / Farina, A; Sciunzi, Berardino; Valdinoci, E.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 7:4(2008), pp. 741-791.
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Titolo: | Bernstein and De Giorgi type problems: new results via a geometric approach |
Autori: | |
Data di pubblicazione: | 2008 |
Rivista: | |
Citazione: | Bernstein and De Giorgi type problems: new results via a geometric approach / Farina, A; Sciunzi, Berardino; Valdinoci, E.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 7:4(2008), pp. 741-791. |
Handle: | http://hdl.handle.net/20.500.11770/132675 |
Appare nelle tipologie: | 1.1 Articolo in rivista |