Periodic homogenisation of Hamilton-Jacobi equations .2. Eikonal equations

Homogenisation of the first-order Hamilton-Jacobi equations u(i)(?)(epsilon) + H(Du(E), x/epsilon) = 0, when H is periodic in the second variable, leads to an effective Hamiltonian (H) over bar satisfying: u(epsilon) converges, as epsilon --> 0, to the solution u of u(t) + (H) over bar(Du)= 0. In our first paper, we assumed that H is convex and we derived a variational formula giving (H) over bar. In this second paper, we consider eikonal equations, i.e. H(p, x)=1/2 \ p \(2) - V(x). Using our variational formula, we compute explicitly the effective Hamiltonian in several cases, and we study precisely the lack of strict convexity for (H) over bar ('flat part' around the origin).