Homogeneous solutions to fully nonlinear elliptic equations

We classify homogeneous degree d 6 2 solutions to fully nonlinear elliptic equations. In this note, we show that any homogeneous degree other than 2 solution to fully nonlinear elliptic equations must be "harmonic". Consider the fully nonlinear elliptic equation F D 2 u � = 0 with µI ≤ (Fij) = FMij (M) � ≤ µ −1 I. Nirenberg (N) derived the a priori C 2,� estimates for the above equa- tion in dimension 2 in 1950s. Krylov (K) and Evans (E) showed the same a priori estimates for the above equations in general dimensions under the assumption that F is convex. As a modest investigation of a priori estimates for general fully nonlinear elliptic equations without convexity condition, we study the homogeneous solutions. Theorem 0.1. Let u be a continuous in R n \ {0} homogeneous degree d 6 2 solution to the elliptic equation F D 2 u � = 0 in R n with F ∈ C 1 . Then u is harmonic in a possible new coordinate system in R n , namely n X i,j=1 Fij (0)Diju(x) = 0.