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

Abstract

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 $u^{\epsilon}$ and the limiting solution $u$. We prove $u^{\epsilon}\rightarrow u$ as $\epsilon \rightarrow 0^+$ qualitatively and quantitatively derive an $\epsilon |\log \epsilon|$ type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples.