Gradient bounds for elliptic problems singular at the boundary

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