On the Γ-limit for a non-uniformly bounded sequence of two-phase metric functionals

We consider the Γ-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either {1, ∞} or {1, βε^-g} where β, ε > 0 and p ∈ (0, ∞). We find that for a large class of metrics, in particular those metrics whose surface of discontinuity forms a differentiable manifold, the Γ-limit exists, as in the case of a uniformly bounded sequence of metrics. However, the existence of the Γ-limit for the corresponding boundary value problem depends on the value of p. Specifically, we show that the power p = 1 is critical in that the Γ-limit exists for p < 1, whereas it ceases to exist for p = 1. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.