Periodic homogenisation of Hamilton–Jacobi equations: 2. Eikonal equations

Homogenisation of the first-order Hamilton-Jacobi equations when H is periodic in the second variable, leads to an effective Hamiltonian H satisfying: uε converges, as ε → 0, to the solution u of ut + H(Du) = 0. In our first paper, we assumed that H is convex and we derived a variational formula giving H. In this second paper, we consider eikonal equations, i.e. H(p, x) = ½|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 (‘flat part’ around the origin).