Bellman Equations with Sub-Lipschitz Hessians

We construct homogeneous solutions with non-Lipschitz Hessian for finite, constant-coefficient Bellman equations. First, for every $\sigma \in (0,1)$, we find two uniformly elliptic matrices $A_1,A_2\in {\mathcal{S}}^4$ and a nonzero $(2+\sigma )$-homogeneous solution $u$ of $$\max \{\mathrm{t}\mathrm{r}(A_1{D}^{2}u),\mathrm{t}\mathrm{r}(A_2{D}^{2}u)\}=0\text{in }{\mathbb{R}}^4\mathrm{.}$$ Second, in ${\mathbb{R}}^2$ we construct three matrices satisfying ${\mathrm{I}\mathrm{d}}_2\le A_j\le 3{\mathrm{I}\mathrm{d}}_2$ for which the corresponding Bellman equation admits a homogeneous solution with a non-Lipschitz Hessian. In particular, solutions to convex fully nonlinear uniformly elliptic equations are not in $C^{2,1}$, and not even in $C^{2,1-\epsilon }$ for $\epsilon >0$ small.