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Linear oblique derivative problems for the uniformly elliptic Hamilton- Jacobi-Bellman equation. (English) Zbl 0593.35046
The Bellman equation which arises in the optimal control of diffusions in a domain with reflecting boundary conditions is the following: $$F[u]\equiv \inf \{L_ ku-f_ k;\quad k\in {\mathbb{N}}\}=0$$ in $$\Omega$$, Mu$$\equiv \beta^ iD_ iu+\gamma u=g$$ on $$\partial \Omega$$, where $$L_ ku\equiv a_ k^{ij} D_{ij}u+b^ i_ k D_ iu+c_ ku$$, the coefficients are real functions on $$\Omega$$ with $$[a_ k^{ij}]$$ positive matrices on $$\Omega$$. Also $$\beta^ i$$, $$1\leq i\leq n$$, and $$\nu$$ are real on $$\partial \Omega$$. M is called oblique if $$\beta \cdot \nu >0$$ on $$\partial \Omega$$ ($$\nu$$ is the inner normal) and regularly oblique if $$\beta \cdot \nu \geq \lambda >0$$ on $$\partial \Omega$$. Of course these operators include the Neumann problem with $$\beta =\nu$$, $$\nu =0.$$
The regularity result proved here is the Theorem: Let $$\Omega$$ be bounded in $${\mathbb{R}}^ n$$ with $$\partial \Omega \in C^{3,1}$$. Suppose F[u] is uniformly elliptic in $$\Omega$$, $$\beta$$ $$\cdot \nu \geq \lambda$$ on $$\partial \Omega$$, all the coefficients in $$L_ k$$ and M and g are in $$C^{1,1}({\bar \Omega})$$ with norms independent of k and $$c_ k$$, $$\nu\leq 0$$, for all k. If $$\sup \{c_ k;k,{\bar \Omega}\}+\sup \{\nu;\partial \Omega \}<0$$ then $$F[u]=0$$ in $$\Omega$$, $$Mu=g$$ on $$\partial \Omega$$ has a unique solution $$u\in C^{1,1}({\bar \Omega})\cap C^{2,\alpha}(\Omega)$$ for some $$\alpha >0$$ depending only on the dimension n and the constants of ellipticity $$\Lambda$$ /$$\lambda$$. Several related results are given and the application to stochastic control is also discussed.
Reviewer: E.Barron

##### MSC:
 35J60 Nonlinear elliptic equations 49L20 Dynamic programming in optimal control and differential games 35B65 Smoothness and regularity of solutions to PDEs
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