# Regularity of linear Bellman equation

Let $$f(x,t):B_1 \times [0,1] \to \mathbb{R}$$ be Lipschitz function on both $$x$$ and $$t$$, $$\varphi$$ be Lipschitz function on both $$x$$ and $$t$$ on the parabolic boundary of $$B_1 \times [0,1]$$. Let $$A$$ be a countable set. Suppose $$a_{ij}^\nu(x,t)$$, $$a_i^\nu(x,t)$$ are regular enough, we consider the linear Bellman equation, $$u_t(x,t) +\sup_{\nu \in A} \Big(a_{ij}^\nu(x,t) u_{x_ix_j}(x,t) +a_i^\nu(x,t) u_{x_i}(x,t) \Big)=f(x,t)$$ on $$B_1 \times [0,1]$$ with parabolic boundary value $$\varphi$$. I am looking for the interior $$C^{2,1}$$ regularity of the solution. I think it is true, but I cannot find a proper reference. There is a close one in Theorem 14.15 "Second Order Parabolic Differential Equations" by Lieberman, Gary M., see also the comments for the linear operator $$L_\nu$$ just below the theorem. It states quite ambiguously in that comment that $$f(x,t)$$ should be differentiable on both $$x,t$$, but I think it is not necessary.

• What is $A$, is it an arbitrary not empty set? Nov 3 at 12:03
• You may think that it is a countable set Nov 4 at 23:44