Global solutions to the eikonal equation

We study structural stability of smoothness of the maximal solution to the geometric eikonal equation on (Rd,G)(Rd,G), d⩾2d⩾2. This is within the framework of order zero metrics G. For a subclass of these metrics we show existence, stability as well as precise asymptotics for derivatives of the solution. These results are applicable to examples arising in Schrödinger operator theory.