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

TL;DR

This paper investigates the $ ext{Gamma}$-limit of highly oscillatory two-phase metric functionals with unbounded coefficients, revealing critical behavior at $p=1$ and implications for nonlinear optics and Hamiltonian systems.

Contribution

It establishes the existence of the $ ext{Gamma}$-limit for certain unbounded metrics and identifies the critical exponent $p=1$ where the limit behavior changes.

Findings

$ ext{Gamma}$-limit exists for $p<1$

Limit does not exist for $p ext{ } ext{geq} ext{ } 1$

Applications in nonlinear optics and Hamiltonian particle models

Abstract

In this study we consider the $\Gamma$-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 $\{1,\beta \varepsilon^{-p}\}$ where $\beta,\varepsilon > 0$ 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 $\Gamma$-limit exists, as in the uniformly bounded case. However, when one attempts to determine the $\Gamma$-limit for the corresponding boundary value problem, the existence of the $\Gamma$-limit depends on the value of $p$. Specifically, we show that the power $p=1$ is critical in that the $\Gamma$-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.