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 $$ {\,{\rm div}\,} \Big |) \nabla u\Big )+f)=0.$$ 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