Effective Hamiltonian and homogenization for measurable Eikonal equations

We study the homogenization problem for a class of evolutive Hamilton–Jacobi equations with measurable dependence on the state variable.We use in our analysis an adaptation of the definition of viscosity solutions to the discontinous setting, obtained through replacement, in the test inequalities, of the punctual value of the Hamiltonian by measure–theoretic weak limits. The existence of periodic solutions to the corresponding cell problem cannot be achieved, under our assumptions, through an ergodic approximation. Instead our approach is based on the introduction of an intrinsic distance defined as the infimum of some line integrals on curves joining two given points. A crucial role for this is played by the notion of transversality between curves and sets of vanishing Lebesgue measure. The asymptotic analysis is finally performed and employs a modified versions of the Evans’ perturbed test-function argument, the most relevant new fact being the use of t-partial sup-convolution and t-partial convolutions of subsolution to compensate the lack of continuity.