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

Alberto Farina; Berardino Sciunzi; Enrico Valdinoci

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 4, page 741-791
  • ISSN: 0391-173X

Abstract

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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  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.

How to cite

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Farina, Alberto, Sciunzi, Berardino, and Valdinoci, Enrico. "Bernstein and De Giorgi type problems: new results via a geometric approach." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.4 (2008): 741-791. <http://eudml.org/doc/272280>.

@article{Farina2008,
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\[ \{\,\{\rm div\}\,\} \Big (a(|\nabla u(x)|) \nabla u(x)\Big )+f(u(x))=0.\]Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in $\mathbb \{R\}^2$ and $\mathbb \{R\}^3$ and of the Bernstein problem on the flatness of minimal area graphs in $\mathbb \{R\}^3$. 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.},
author = {Farina, Alberto, Sciunzi, Berardino, Valdinoci, Enrico},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Bernstein problem; De Giorgi problem; quasilinear elliptic equation},
language = {eng},
number = {4},
pages = {741-791},
publisher = {Scuola Normale Superiore, Pisa},
title = {Bernstein and De Giorgi type problems: new results via a geometric approach},
url = {http://eudml.org/doc/272280},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Farina, Alberto
AU - Sciunzi, Berardino
AU - Valdinoci, Enrico
TI - Bernstein and De Giorgi type problems: new results via a geometric approach
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 4
SP - 741
EP - 791
AB - 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\[ {\,{\rm div}\,} \Big (a(|\nabla u(x)|) \nabla u(x)\Big )+f(u(x))=0.\]Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in $\mathbb {R}^2$ and $\mathbb {R}^3$ and of the Bernstein problem on the flatness of minimal area graphs in $\mathbb {R}^3$. 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.
LA - eng
KW - Bernstein problem; De Giorgi problem; quasilinear elliptic equation
UR - http://eudml.org/doc/272280
ER -

References

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