Consider the eikonal equation $$ \left\lbrace \begin{array}{lcl} ||\nabla\phi|| = f(x) &, & x \in \Omega \\ \phi(x) = 0 &, & x \in \Gamma \subseteq \partial \Omega \end{array} \right. $$
Is there any way to deduce, only from that PDE, a variational principle similar to Fermat's principle in Geometric Optics? I mean, is it posible to conclude that the line integral $$ \phi[x(s)] = \int f(x(s)) \ ds $$ reaches its minimum along the light rays (characteristic curves of the PDE)?
I've tried to apply the ODEs the method of characteristics provide when applied to the eikonal equation, but with no conclusive results.
Thanks in advance.