Classical global solutions for a class of Hamilton-Jacobi equations (Q1958733)

Let \(V\) be a real-valued \(C^2\)-function on \(\mathbb R^n\), \(n\geq 2\), such that \(V(x)=0\) for \(|x|\leq R\) and \(\partial_x^\alpha V(x)=O(|x|^{-\epsilon-|\alpha|})\) (as \(|x|\to\infty\)) for some \(\epsilon>0\) and all multiindices with \(|\alpha|\leq 2\). Using the method of characteristics and a global inverse function theorem of Hadamard the authors shows that, for sufficiently large values of \(R>0\), the Hamilton-Jacobi equation of eikonal type, \[ |\nabla u(x)|^2+V(x) = k^2,\quad k>0, \] has a \(C^1\)-solution on \(\mathbb R^n\setminus\{0\}\).