# American Institute of Mathematical Sciences

January  2015, 35(1): 411-426. doi: 10.3934/dcds.2015.35.411

## 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, United Kingdom, United Kingdom 2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received  January 2014 Revised  June 2014 Published  August 2014

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.
Citation: Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
##### References:
 [1] E. Acerbi and G. Buttazzo, On the limits of periodic Riemannian metrics,, J. Analyse Math., 43 (): 183. doi: 10.1007/BF02790183. Google Scholar [2] M. Amar, G. Crasta and A. Malusa, On the Finsler metric obtained as limits of chessboard structures,, Adv. Calc. Var., 2 (2009), 321. doi: 10.1515/ACV.2009.013. Google Scholar [3] M. Amar and E. Vitali, Homogenization of periodic Finsler metrics,, J. Convex Anal., 5 (1998), 171. Google Scholar [4] V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4757-2063-1. Google Scholar [5] A. Banerjee and N. Adams, Dynamics of classical systems based on the principle of stationary action,, J. Chem. Phys., 92 (): 7330. doi: 10.1063/1.458218. Google Scholar [6] A. Braides, Almost periodic methods in the theory of homogenization,, Appl. Anal., 47 (1992), 259. doi: 10.1080/00036819208840144. Google Scholar [7] A. Braides, $\Gamma$-convergence for Beginners,, Oxford University Press, (2002). doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar [8] A. Braides, G. Buttazzo and I. Fragalá, Riemannian approximation of Finsler metrics,, Asymptotic Analysis, 31 (2002), 177. Google Scholar [9] A. Braides and A. Defranceschi, Homogenisation of Multiple Integrals,, Oxford University Press, (1998). Google Scholar [10] D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry,, American Mathematical Society, (2001). Google Scholar [11] G. Buttazzo, L. D. Pascale and I. Fragalá, Topological equivalence of some variational problems involving distances,, Discrete and Continuous Dynamical Systems, 7 (2001), 247. doi: 10.3934/dcds.2001.7.247. Google Scholar [12] M. C. Concordel, Periodic homogenisation of Hamilton-Jacobi equations. II. Eikonal equations,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 665. doi: 10.1017/S0308210500023763. Google Scholar [13] B. Craciun and K. Bhattacharya, Homogenisation of a Hamilton-Jacobi equation associated with the geometric motion of an interface,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773. doi: 10.1017/S0308210500002675. Google Scholar [14] W. E, A class of homogenisation problems in the calculus of variations,, Comm. Pure Appl. Math., 44 (1991), 733. doi: 10.1002/cpa.3160440702. Google Scholar [15] J. Jost, Riemannian Geometry and Geometric Analysis,, 4th edition, (2005). Google Scholar [16] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenisation of Hamilton-Jacobi equations,, preprint, (1988). Google Scholar [17] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,, Springer-Verlag, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [18] A. Oberman, R. Takei and A. Vladimirsky, Homogenisation of metric Hamilton-Jacobi equations,, Multiscale Model. Simul., 8 (2009), 269. doi: 10.1137/080743019. Google Scholar [19] H. Schwetlick, D. C. Sutton and J. Zimmer, The Finsler metric obtained as the $\Gamma$-limit of a generalised Manhattan metric,, to appear in J. Convex Anal., (). Google Scholar [20] H. Schwetlick and J. Zimmer, Calculation of long time classical trajectories: Algorithmic treatment and applications for molecular systems,, J. Chem. Phys., 130 (2009). doi: 10.1063/1.3096294. Google Scholar [21] D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description,, Ph.D thesis, (2013). Google Scholar

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##### References:
 [1] E. Acerbi and G. Buttazzo, On the limits of periodic Riemannian metrics,, J. Analyse Math., 43 (): 183. doi: 10.1007/BF02790183. Google Scholar [2] M. Amar, G. Crasta and A. Malusa, On the Finsler metric obtained as limits of chessboard structures,, Adv. Calc. Var., 2 (2009), 321. doi: 10.1515/ACV.2009.013. Google Scholar [3] M. Amar and E. Vitali, Homogenization of periodic Finsler metrics,, J. Convex Anal., 5 (1998), 171. Google Scholar [4] V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4757-2063-1. Google Scholar [5] A. Banerjee and N. Adams, Dynamics of classical systems based on the principle of stationary action,, J. Chem. Phys., 92 (): 7330. doi: 10.1063/1.458218. Google Scholar [6] A. Braides, Almost periodic methods in the theory of homogenization,, Appl. Anal., 47 (1992), 259. doi: 10.1080/00036819208840144. Google Scholar [7] A. Braides, $\Gamma$-convergence for Beginners,, Oxford University Press, (2002). doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar [8] A. Braides, G. Buttazzo and I. Fragalá, Riemannian approximation of Finsler metrics,, Asymptotic Analysis, 31 (2002), 177. Google Scholar [9] A. Braides and A. Defranceschi, Homogenisation of Multiple Integrals,, Oxford University Press, (1998). Google Scholar [10] D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry,, American Mathematical Society, (2001). Google Scholar [11] G. Buttazzo, L. D. Pascale and I. Fragalá, Topological equivalence of some variational problems involving distances,, Discrete and Continuous Dynamical Systems, 7 (2001), 247. doi: 10.3934/dcds.2001.7.247. Google Scholar [12] M. C. Concordel, Periodic homogenisation of Hamilton-Jacobi equations. II. Eikonal equations,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 665. doi: 10.1017/S0308210500023763. Google Scholar [13] B. Craciun and K. Bhattacharya, Homogenisation of a Hamilton-Jacobi equation associated with the geometric motion of an interface,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773. doi: 10.1017/S0308210500002675. Google Scholar [14] W. E, A class of homogenisation problems in the calculus of variations,, Comm. Pure Appl. Math., 44 (1991), 733. doi: 10.1002/cpa.3160440702. Google Scholar [15] J. Jost, Riemannian Geometry and Geometric Analysis,, 4th edition, (2005). Google Scholar [16] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenisation of Hamilton-Jacobi equations,, preprint, (1988). Google Scholar [17] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,, Springer-Verlag, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [18] A. Oberman, R. Takei and A. Vladimirsky, Homogenisation of metric Hamilton-Jacobi equations,, Multiscale Model. Simul., 8 (2009), 269. doi: 10.1137/080743019. Google Scholar [19] H. Schwetlick, D. C. Sutton and J. Zimmer, The Finsler metric obtained as the $\Gamma$-limit of a generalised Manhattan metric,, to appear in J. Convex Anal., (). Google Scholar [20] H. Schwetlick and J. Zimmer, Calculation of long time classical trajectories: Algorithmic treatment and applications for molecular systems,, J. Chem. Phys., 130 (2009). doi: 10.1063/1.3096294. Google Scholar [21] D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description,, Ph.D thesis, (2013). Google Scholar
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