Let Ω be a bounded smooth domain in
\({{R}^N, N \geqq 2}\), and let us denote by
d(
x) the distance function
d(
x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model is
$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $
where
f belongs to
\({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at
\({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field
\({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies
\({B\cdot \nu\geqq \alpha}\) at ?Ω (
ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to
α, as
α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.