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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Control of degenerate diffusions in $ {\bf R}\sp d$


Author: Omar Hijab
Journal: Trans. Amer. Math. Soc. 327 (1991), 427-448
MSC: Primary 35B65; Secondary 35B37, 35J60, 93C20
MathSciNet review: 1040262
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Abstract: An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation

$\displaystyle - Av + F(x,\upsilon ,D\upsilon )= 0,$

$ 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1040262-2
PII: S 0002-9947(1991)1040262-2
Keywords: Nonlinear partial differential equation, diffusion, control, dynamic programming
Article copyright: © Copyright 1991 American Mathematical Society