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.

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