Maximal generalized solution of eikonal equation

We study the Dirichlet problem for the eikonal equation:{12|∇u(x)|2−a(x)=0inΩu(x)=φ(x)on∂Ω, without continuity assumptions on the map a(⋅)a(⋅). We find a class of maps a(⋅)a(⋅) contained in the space L∞(Ω)L∞(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.