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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

Hartmut Schwetlick 1, , Daniel C. Sutton 1, and  Johannes Zimmer 2,

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.
Keywords: homogenisation, differential geometry, Maupertuis principle., $\Gamma$-convergence, Hamiltonian dynamics.
Mathematics Subject Classification: 49J45, 53C6.
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]

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[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

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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

show all references

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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