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

In this study 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 $\{1,\infty\}$ or $\{1,β\varepsilon^{-p}\}$ where $β,\varepsilon > 0$ and $p \in (0,\infty)$. 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 uniformly bounded case. However, when one attempts to determine the $Γ$-limit for the corresponding boundary value problem, the existence of the $Γ$-limit 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 $p \geq 1$. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.