The vanishing viscosity process for an eikonal equation in the radially symmetric setting

We study the vanishing viscosity method for the eikonal equation $|Du|=V$ in $B(0,1)$ with homogeneous Dirichlet boundary value condition. By assuming $V$ is radially symmetric and restricting attention to radially symmetric solutions, we construct explicit formulas for both the viscous solution $u^ε$ and the limiting solution $u$. We prove $u^ε\rightarrow u$ as $ε\rightarrow 0^+$ qualitatively and quantitatively derive an $ε|\log ε|$ type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples.