Radial symmetry of classical solutions for Bellman equations in ergodic control (Q2712231)

Without imposing Lyapunov-type stability conditions, the authors study the \(d\)-dimensional Bellman equation NEWLINE\[NEWLINE\lambda=\frac 12\Delta \varphi- \|\nabla\varphi \|+h,NEWLINE\]NEWLINE where \(h\) is a convex function of polynomial growth, and \(\lambda\), \(\varphi\) are an unknown constant and \(C^2\)-function, respectively. The existence of a unique classical solution is shown via the vanishing discount approach. If \(h\) is radial, i.e. \(h(x)=f(\|x\|)\), the equation ``inherits'' the radial symmetry, an explicit solution is possible, and the optimal control is given by a feedback law.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].