Symmetry of Global Solutions to a Class of Fully Nonlinear Elliptic Equations in 2D

摘要

We prove that entire bounded monotone solutions to fully nonlinear equations in 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 = |¤ and f(u) = u~3 - u, our result gives a new (non-variational) proof of the well known De Giorgi's conjecture.