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Existence results for Bellman equations and maximum principles in unbounded domains. (English) Zbl 0961.35021

The Hamilton-Jacobi-Bellman equation: \[ \sup \limits_{k \in \mathbb{N}} (L^k u(x) + f^k(x)) = 0 \quad \text{in } \Omega, \quad u = 0 \quad \text{on }\partial\Omega, \] where \(L^k\) is a countable family of uniformly elliptic operators in the form \[ L^k=\sum_{1\leq i,j \leq p} a_{ij}^k(y) \partial_i\partial_j+ \sum_{1\leq i \leq p}b_{i}^k(y)\partial_{i}+ c^k(y), \] and \(\Omega\) is a bounded \(C^{2,\alpha}\) domain in \(\mathbb{R}^p\), is considered. An existence theorem is proved in case when the principal eigenvalues of the operators \(L^k\) are bounded from below by a positive constant. Under the condition that the coefficients of \(L^k\) are from the class \(C^{0,\alpha}\), an a priori \(C^{2,\gamma}\) estimate of the solution is established. The maximum principle in the infinite small cross-section cylinder \(\Omega\times \mathbb{R}^q= \{(y,z)\mid y \in \Omega\), \(z\in \mathbb{R}^q\}\) for the semilinear elliptic equation was established earlier if the leading coefficients depended on \(y\) only. This result is transfered to the case where these coefficients depend on \(y\) and \(z\). The maximum principle is used to prove the theorem of symmetry for the solution with respect to the hyperplane \(y_1=0\).

MSC:

35B50 Maximum principles in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B45 A priori estimates in context of PDEs
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