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→ u as ε→ 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.