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

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

$$\mathrm{div}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$

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 very degenerate operators: as an application, we prove one-dimensional symmetry for $1$-Laplacian type operators.