Geometrical Properties of Fully Nonlinear Equations and an Application to Singularities

In this paper several results on geometrical properties of viscosity solutions of fully nonlinear equations are presented. In particular I prove symmetry and monotonicity for solutions of fully nonlinear second order equations defined in finite cylinders, and asymptotic symmetry for solutions of fully nonlinear second order equations defined in exterior domains. Also I utilize such geometrical ideas to study the asymptotic behavior of singular viscosity solutions of first order Hamilton-Jacobi equations.