Hausdorff Continuous Viscosity Solutions of Hamilton-Jacobi Equations
  A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the Hamilton-Jacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions can be extended to the wider space of Hausdorff continuous functions while also generalizing some of the existing concepts of discontinuous solutions. 
Roumen Anguelov and Froduald Minani