Radial Symmetry of Classical Solutions for Bellman Equations in Ergodic Control

We are concerned with the d-dimensional Bellman equation of the form: $$\lambda = \frac{1}{2}\Delta \phi (x) + F(D\phi (x)) + h(x),\quad x \in {R^d},$$ (1.1) where $$F(\xi ) = \min \{ (\xi ,p):|p| \leqslant \} = - |\xi |,$$ (1.2) and │·│, (,), and D denote the norm, the inner product of vectors, and the gradient respectively. We are given a convex function h(x) with polynomial growth, and the unknown is the pair of a constant λ and a C 2-function φ(x) on R d.