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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

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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