Analysis and Approximation of Hamilton Jacobi equations on irregular data

We will present some numerical schemes for some non classical Hamilton-Jacobi equations. We will consider an Eikonal equation with discontinuous coefficients. In this case, through a particular condition on the discontinuities we can preserve a comparison principle for the solutions. We will introduce ans study a semiLagrangian numerical scheme also deriving error bounds. Another case which we will present is an eikonal equation on a graph. We will briefly present the notion of viscosity solution in this situation introduced by Camilli and al. in 2010 and we will build a numerical approximation for it. Some numerical solutions for the Shape-from-Shading problem and for an optimal control problem on a domain with constraints will be presented.