Control of degenerate diffusions in $\textbf {R}^ d$

Abstract:

An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation \[ - Av + F(x,\upsilon ,D\upsilon )= 0,\] $A= \frac {1}{2}\sum {{a_{ij}}(x){\partial ^2}/\partial x{_i} \partial {x_j}}, x \in {{\mathbf {R}}^d}$. We assume the equation is of Bellman type, i.e. $F(x,\upsilon ,p)= {\sup _{u \in U}}[b(x,u) \cdot p + c(x,u)\upsilon - f(x,u)]$, $U \subset {{\mathbf {R}}^d}$. If we set $\lambda \equiv {\inf _{x,u}}c(x,u)$, then there exists ${\lambda _0} \geq 0$ such that $0 < \lambda < {\lambda _0}$ implies $\upsilon$ is Hölder, while $\lambda > {\lambda _0}$ implies $\upsilon$ is Lipschitz. The following is established: Suppose the equation is also of Lipschitz type, i.e. suppose there is a Lipschitz function $u(x,\upsilon ,p)$ such that the supremum in $F (x,\upsilon ,p)$ is uniquely attained at $u= u (x,\upsilon ,p)$; then there exists ${\lambda _1} > {\lambda _0}$ such that $\lambda > {\lambda _1}$ implies $\upsilon$ is ${C^{1,1}},$ i.e. $D\upsilon$ exists and is Lipschitz.