Summary: Bellman systems \(-\Delta u+ \alpha u= H(.,\nabla u)\), \(u\in H^{1,2}_0({\mathcal O},\mathbb{R}^3)\), \({\mathcal O}\) a bounded domain of \(\mathbb{R}^n\) are considered. The Hamiltonian has quadratic growth in \(\nabla u\). The technique to obtain estimates in \(H^1({\mathcal O})\), \(C^\alpha({\mathcal O})\) and \(H^{2,q}({\mathcal O})\) for solutions \(u\) is explained for the case that \(H\) satisfies a structure condition of the following type: \[ |H_\nu(p)|\leq K|p||p_\nu|+ \sum_{\mu\leq\nu} |p_\nu|^2+ K \] . Simple stochastic differential games related to the Bellman system are considered.
For the entire collection see [Zbl 1053.49001].