# Evans-Krylov theorem

The Evans-Krylov theorem says that if $u$ is a solution to a uniformly elliptic, fully nonlinear, convex or concave, equation $F(D^2 u) = 0 \text{ in } B_1,$ then $u \in C^{2,\alpha}(B_{1/2})$ and there is an estimate $\|u\|_{C^{2,\alpha}(B_{1/2})} \leq C \|u\|_{L^\infty(B_1)},$ where $C$ and $\alpha>0$ depend only on the ellipticity constants and dimension.