In this work we present and efficient approach to homogenization for a class of static Hamilton-Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized Hamiltonian. This is significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piece-wise constant periodic and random speed functions.

We will also discuss some recent results on homogenization of second order fully nonlinear equations.