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

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci

Research output: Contribution to journalArticle

100 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)741-791
Number of pages51
JournalAnnali della Scuola Normale - Classe di Scienze
Volume7
Issue number4
Publication statusPublished - 1 Dec 2008
Externally publishedYes

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Geometric Approach
Symmetry
Singular Elliptic Equation
Degenerate Elliptic Equations
Stable Solution
Flatness
Operator
Level Set
Half-space
Byproducts
Phase Transition
Phase transitions
Graph in graph theory

Cite this

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Bernstein and De Giorgi type problems : New results via a geometric approach. / Farina, Alberto; Sciunzi, Berardino; Valdinoci, Enrico.

In: Annali della Scuola Normale - Classe di Scienze, Vol. 7, No. 4, 01.12.2008, p. 741-791.

Research output: Contribution to journalArticle

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AU - Sciunzi, Berardino

AU - Valdinoci, Enrico

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