Below is the ascii version of the abstract for 07-171. The html version should be ready soon.

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci
Bernstein and De Giorgi type
problems: new results via a geometric approach
(434K, pdf)

ABSTRACT.  We use a Poincar\'e 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.