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.