The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint

We consider solutions of the equation -Delta u + lambda u + |Delta u|(q) = f, which blow up uniformly at the boundary of a smooth domain, that can be interpreted as the value function of a state constraint control problem for a Brownian motion. We prove a complete asymptotic expansion of the gradient at the boundary, giving the precise behavior of normal and tangent components. The result is achieved by proving Lipschitz regularity for u - S, where S is an explicit singular corrector term. As the main motivation and application of our result, we characterize the behavior of the singular optimal control law and of the constrained dynamics near the boundary.