American Institute of Mathematical Sciences

Advanced
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, 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), 321-360. doi: 10.1515/ACV.2009.013.  Google Scholar

[3]

M. Amar and E. Vitali, Homogenization of periodic Finsler metrics, J. Convex Anal., 5 (1998), 171-186.  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), 259-277. 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), 177-187.  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), 247-258. 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), 665-689. 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), 773-805. 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), 733-759. doi: 10.1002/cpa.3160440702.  Google Scholar

[15]

J. Jost, Riemannian Geometry and Geometric Analysis, 4th edition, Universitext, Springer-Verlag, Berlin, 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, New York, 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), 269-295. 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), 124106. doi: 10.1063/1.3096294.  Google Scholar

[21]

D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description, Ph.D thesis, University of Bath, 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), 321-360. doi: 10.1515/ACV.2009.013.  Google Scholar

[3]

M. Amar and E. Vitali, Homogenization of periodic Finsler metrics, J. Convex Anal., 5 (1998), 171-186.  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), 259-277. 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), 177-187.  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), 247-258. 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), 665-689. 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), 773-805. 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), 733-759. doi: 10.1002/cpa.3160440702.  Google Scholar

[15]

J. Jost, Riemannian Geometry and Geometric Analysis, 4th edition, Universitext, Springer-Verlag, Berlin, 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, New York, 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), 269-295. 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), 124106. doi: 10.1063/1.3096294.  Google Scholar

[21]

D. C. Sutton, Macroscopic Hamiltonian Systems and Their Effective Description, Ph.D thesis, University of Bath, 2013. Google Scholar

[1]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078
[2]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[3]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[4]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
[5]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[6]

Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355
[7]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[8]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
[9]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[10]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[11]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427
[12]

Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059
[13]

Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017
[14]

Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605
[15]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[16]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437
[17]

Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223
[18]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[19]

Gideon Simpson, Michael I. Weinstein, Philip Rosenau. On a Hamiltonian PDE arising in magma dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 903-924. doi: 10.3934/dcdsb.2008.10.903
[20]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences