Maximal generalized solution of eikonal equation

We study the Dirichlet problem for the eikonal equation: egin{displaymath} egin{cases} ha | u(x)|^2 - a(x) = 0 & extrm{in} Ocr u(x)=arphi(x) & extrm{on} partial O, end{cases} end{displaymath} without continuity assumptions on the map $a(cdot)$. We find a class of maps $a(cdot)$'s contained in the space $L^infty(O)$ for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution