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

TL;DR

This paper analyzes the vanishing viscosity method for a radially symmetric eikonal equation, providing explicit solutions, convergence rates, and discussing solution uniqueness.

Contribution

It constructs explicit formulas for viscous and limiting solutions under radial symmetry and establishes convergence rates as viscosity vanishes.

Findings

Explicit formulas for viscous and limiting solutions

Quantitative convergence rate of psilon |log psilon

Discussion on uniqueness of viscosity solutions

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.