On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Dolbeault, Jean; Monneau, Régis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 181-197.

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in ${ℝ}^2$. We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

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