Bellman Equations with Sub-Lipschitz Hessians

Abstract

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 \[\max\bigl\{{\rm tr}\,(A_1D^2u),{\rm tr}\,(A_2D^2u)\bigr\}=0 \qquad\text{in }\mathbb{R}^4.\] Second, in $\mathbb{R}^2$ we construct three matrices satisfying ${\rm Id}_2\leq A_j\leq3{\rm Id}_2$ 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 $C^{2, 1-\varepsilon}$ for $\varepsilon > 0$ small.