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

created by farina on 28 Nov 2007
modified on 12 May 2011

[BibTeX]

Published Paper

Inserted: 28 nov 2007
Last Updated: 12 may 2011

Journal: Annali della Scuola Normale di Pisa
Year: 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 equation 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 $\R^2$ 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 non-elliptic operators: as an application, we prove one-dimensional symmetry for $1$-Laplacian type operators.