article

Title: Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D

Authors: D. De Silva and O. Savin

Issue: Volume 58 (2009), Issue 1, 301-316

Abstract:

$We prove that entire bounded monotone solutions to fully nonlinear equations in $\mathbb{R}^2$ of the form $F(D^2u) = f(u)$ are one-dimensional, under appropriate compatibility conditions for $F$ and $f$. In the particular case when $F = \Delta$ and $f(u) = u^3-u$, our result gives a new (non-variational) proof of the well known De Giorgi's conjecture.$