On the $\Gamma$limit for a nonuniformly bounded sequence of twophase metric functionals
Hartmut Schwetlick  Department of Mathematical Sciences, The University of Bath, Bath, BA2 7AY, United Kingdom (email) Abstract: 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 $\{1,\beta \epsilon^{p}\}$ where $\beta,\epsilon > 0$ 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 case of a uniformly bounded sequence of metrics. However, the existence of the $\Gamma$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 $\Gamma$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.
Keywords: $\Gamma$convergence, homogenisation, differential geometry, Hamiltonian dynamics, Maupertuis principle.
Received: January 2014; Revised: June 2014; Available Online: August 2014. 
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