Classical Global Solutions for a Class of Hamilton–Jacobi Equations

Abstract

Let V be a real-valued function of class C ² on ${\mathbb{R}^n}$, n ≥ 2, which vanishes if |x| ≤ R and, for some ${\epsilon >0 }$, satisfies ${\partial_x^\alpha V(x)=O(|x|^{-\epsilon-|\alpha|}) }$, as |x| → ∞, for |α| ≤ 2. Using a global inverse function theorem of Hadamard, we show that if R is sufficiently large, then the Hamilton–Jacobi equation of eikonal type |∇u|² + V(x) = k ², with k > 0, has a C ¹ solution on ${\mathbb{R}^n\setminus\{0\}}$.