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

We study the vanishing viscosity method for the eikonal equation $\mathrm{\mid }Du\mathrm{\mid }=V$ 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 $u^{\epsilon }$ and the limiting solution $u$. We prove $u^{\epsilon }\to u$ as $\epsilon \to 0^{+}$ qualitatively and quantitatively derive an $\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid }$ type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples.