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

Hartmut Schwetlick; Daniel C. Sutton; Johannes Zimmer

doi:10.3934/dcds.2015.35.411

Article Contents

2015, Volume 35, Issue 1: 411-426. Doi: 10.3934/dcds.2015.35.411

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

1.
Department of Mathematical Sciences, The University of Bath, Bath, BA2 7AY
2.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received: December 31, 2013

Revised: May 31, 2014

Published: December 2014

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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.

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Mathematics Subject Classification: 49J45, 53C60.

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