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

We use a Poincare type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div(a(vertical bar del u(x)vertical bar)del u(x)) + 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 very degenerate operators: as an application, we prove one-dimensional symmetry for 1-Laplacian type operators.