### Abstract

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, ∞} or {1, βε^{-g}} where β, ε > 0 and p ∈ (0, ∞). 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 case of a uniformly bounded sequence of metrics. However, the existence of the Γ-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 Γ-limit exists for p < 1, whereas it ceases to exist for p = 1. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.

Original language | English |
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Pages (from-to) | 411-426 |

Number of pages | 16 |

Journal | Discrete and Continuous Dynamical Systems - Series A |

Volume | 35 |

Issue number | 1 |

Early online date | 1 Aug 2014 |

DOIs | |

Publication status | Published - Jan 2015 |

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**On the Γ-limit for a non-uniformly bounded sequence of two-phase metric functionals.** / Schwetlick, Harmut; Sutton, Daniel C.; Zimmer, Johannes.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series A*, vol. 35, no. 1, pp. 411-426. https://doi.org/10.3934/dcds.2015.35.411

}

TY - JOUR

T1 - On the Γ-limit for a non-uniformly bounded sequence of two-phase metric functionals

AU - Schwetlick, Harmut

AU - Sutton, Daniel C.

AU - Zimmer, Johannes

PY - 2015/1

Y1 - 2015/1

N2 - 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, ∞} or {1, βε-g} where β, ε > 0 and p ∈ (0, ∞). 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 case of a uniformly bounded sequence of metrics. However, the existence of the Γ-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 Γ-limit exists for p < 1, whereas it ceases to exist for p = 1. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.

AB - 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, ∞} or {1, βε-g} where β, ε > 0 and p ∈ (0, ∞). 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 case of a uniformly bounded sequence of metrics. However, the existence of the Γ-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 Γ-limit exists for p < 1, whereas it ceases to exist for p = 1. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.

UR - http://www.scopus.com/inward/record.url?scp=84907185254&partnerID=8YFLogxK

UR - http://dx.doi.org/10.3934/dcds.2015.35.411

U2 - 10.3934/dcds.2015.35.411

DO - 10.3934/dcds.2015.35.411

M3 - Article

VL - 35

SP - 411

EP - 426

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 1

ER -