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

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 div (a( ∇u(x) )∇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 ℝ2 and ℝ 3 and of the Bernstein problem on the flatness of minimal area graphs in ℝ3. 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.