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

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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

Abstract
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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 and Continuous Dynamical Systems, 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.
[2]	M. Amar, G. Crasta and A. Malusa, On the Finsler metric obtained as limits of chessboard structures, Adv. Calc. Var., 2 (2009), 321-360. doi: 10.1515/ACV.2009.013.
[3]	M. Amar and E. Vitali, Homogenization of periodic Finsler metrics, J. Convex Anal., 5 (1998), 171-186.
[4]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1997. doi: 10.1007/978-1-4757-2063-1.
[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.
[6]	A. Braides, Almost periodic methods in the theory of homogenization, Appl. Anal., 47 (1992), 259-277. doi: 10.1080/00036819208840144.
[7]	A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[8]	A. Braides, G. Buttazzo and I. Fragalá, Riemannian approximation of Finsler metrics, Asymptotic Analysis, 31 (2002), 177-187.
[9]	A. Braides and A. Defranceschi, Homogenisation of Multiple Integrals, Oxford University Press, 1998.
[10]	D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, American Mathematical Society, 2001.
[11]	G. Buttazzo, L. D. Pascale and I. Fragalá, Topological equivalence of some variational problems involving distances, Discrete and Continuous Dynamical Systems, 7 (2001), 247-258. doi: 10.3934/dcds.2001.7.247.
[12]	M. C. Concordel, Periodic homogenisation of Hamilton-Jacobi equations. II. Eikonal equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 665-689. doi: 10.1017/S0308210500023763.
[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), 773-805. doi: 10.1017/S0308210500002675.
[14]	W. E, A class of homogenisation problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759. doi: 10.1002/cpa.3160440702.
[15]	J. Jost, Riemannian Geometry and Geometric Analysis, 4th edition, Universitext, Springer-Verlag, Berlin, 2005.
[16]	P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenisation of Hamilton-Jacobi equations, preprint, 1988.
[17]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[18]	A. Oberman, R. Takei and A. Vladimirsky, Homogenisation of metric Hamilton-Jacobi equations, Multiscale Model. Simul., 8 (2009), 269-295. doi: 10.1137/080743019.
[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., ().
[20]	H. Schwetlick and J. Zimmer, Calculation of long time classical trajectories: Algorithmic treatment and applications for molecular systems, J. Chem. Phys., 130 (2009), 124106. doi: 10.1063/1.3096294.
[21]	D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description, Ph.D thesis, University of Bath, 2013.

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.
[2]	M. Amar, G. Crasta and A. Malusa, On the Finsler metric obtained as limits of chessboard structures, Adv. Calc. Var., 2 (2009), 321-360. doi: 10.1515/ACV.2009.013.
[3]	M. Amar and E. Vitali, Homogenization of periodic Finsler metrics, J. Convex Anal., 5 (1998), 171-186.
[4]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1997. doi: 10.1007/978-1-4757-2063-1.
[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.
[6]	A. Braides, Almost periodic methods in the theory of homogenization, Appl. Anal., 47 (1992), 259-277. doi: 10.1080/00036819208840144.
[7]	A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[8]	A. Braides, G. Buttazzo and I. Fragalá, Riemannian approximation of Finsler metrics, Asymptotic Analysis, 31 (2002), 177-187.
[9]	A. Braides and A. Defranceschi, Homogenisation of Multiple Integrals, Oxford University Press, 1998.
[10]	D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, American Mathematical Society, 2001.
[11]	G. Buttazzo, L. D. Pascale and I. Fragalá, Topological equivalence of some variational problems involving distances, Discrete and Continuous Dynamical Systems, 7 (2001), 247-258. doi: 10.3934/dcds.2001.7.247.
[12]	M. C. Concordel, Periodic homogenisation of Hamilton-Jacobi equations. II. Eikonal equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 665-689. doi: 10.1017/S0308210500023763.
[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), 773-805. doi: 10.1017/S0308210500002675.
[14]	W. E, A class of homogenisation problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759. doi: 10.1002/cpa.3160440702.
[15]	J. Jost, Riemannian Geometry and Geometric Analysis, 4th edition, Universitext, Springer-Verlag, Berlin, 2005.
[16]	P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenisation of Hamilton-Jacobi equations, preprint, 1988.
[17]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[18]	A. Oberman, R. Takei and A. Vladimirsky, Homogenisation of metric Hamilton-Jacobi equations, Multiscale Model. Simul., 8 (2009), 269-295. doi: 10.1137/080743019.
[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., ().
[20]	H. Schwetlick and J. Zimmer, Calculation of long time classical trajectories: Algorithmic treatment and applications for molecular systems, J. Chem. Phys., 130 (2009), 124106. doi: 10.1063/1.3096294.
[21]	D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description, Ph.D thesis, University of Bath, 2013.

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