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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 411 - 426, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.411

 
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Hartmut Schwetlick - Department of Mathematical Sciences, The University of Bath, Bath, BA2 7AY, United Kingdom (email)
Daniel C. Sutton - Department of Mathematical Sciences, The University of Bath, Bath, BA2 7AY, United Kingdom (email)
Johannes Zimmer - Department of Mathematical Sciences, 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.
Mathematics Subject Classification:  49J45, 53C60.

Received: January 2014;      Revised: June 2014;      Available Online: August 2014.

 References