TY - JOUR

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

AU - Schwetlick, Hartmut

AU - Sutton, Daniel C.

AU - Zimmer, Johannes

N1 - Publisher Copyright:
© 2015 AIMS Sciences. All rights reserved.

PY - 2015/1/1

Y1 - 2015/1/1

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.

KW - Differential geometry

KW - Hamiltonian dynamics

KW - Homogenisation

KW - Maupertuis principle

KW - Γ-convergence

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

U2 - 10.3934/dcds.2015.35.411

DO - 10.3934/dcds.2015.35.411

M3 - Article

AN - SCOPUS:84907185254

SN - 1078-0947

VL - 35

SP - 411

EP - 426

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 1

ER -