07-171 Alberto Farina, Berardino Sciunzi, Enrico Valdinoci
Bernstein and De Giorgi type problems: new results via a geometric approach (434K, pdf) Jul 4, 07
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Abstract. We use a Poincar\'e 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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