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 ℝ2 and ℝ 3 and of the Bernstein problem on the flatness of minimal area graphs in ℝ3. 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 / A. Farina, B. Sciunzi, E. Valdinoci. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 7:4(2008), pp. 741-791.

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

E. Valdinoci
2008

Abstract

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 ℝ2 and ℝ 3 and of the Bernstein problem on the flatness of minimal area graphs in ℝ3. 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.
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/266028
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