Entire solutions to a class of fully nonlinear elliptic equations

Savin, Ovidiu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, pp. 369-405.

We study nonlinear elliptic equations of the form $F\left(D^{2}u\right)=f\left(u\right)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $\xi \in {ℝ}^n$. We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

Classification : 35J70, 35B65

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