April  2022, 17(2): 227-254. doi: 10.3934/nhm.2022004

Stochastic two-scale convergence and Young measures

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

2. 

Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching bei München, Germany

3. 

Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany

*Corresponding author: Stefan Neukamm

Received  June 2021 Revised  January 2022 Published  April 2022 Early access  February 2022

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.

Citation: Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

K. T. Andrews and S. Wright, Stochastic homogenization of elliptic boundary-value problems with $L^p$-data, Asymptot. Anal., 17 (1998), 165-184. 

[3]

T. ArbogastJ. Douglas, Jr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[4]

E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), 570-598.  doi: 10.1137/0322035.

[5]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flow, Comptes Rendusa l'Académie des Sciences, 320 (1994), 1289–1294.

[6]

A. BourgeatA. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. 

[7]

D. CioranescuA. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math., 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028.

[8]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[9]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math., 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[10]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization., Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[12]

D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics. Springer-Verlag, New York, 1988.

[13]

J. Fischer and S. Neukamm, Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Arch. Ration. Mech. Anal., 242 (2021), 343-452.  doi: 10.1007/s00205-021-01686-9.

[14]

G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. 

[15]

H. Hanke, Homogenization in gradient plasticity, Math. Models Methods Appl. Sci., 21 (2011), 1651-1684.  doi: 10.1142/S0218202511005520.

[16]

M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022.

[17]

M. Heida, Stochastic homogenization of rate-independent systems and applications, Contin. Mech. Thermodyn., 29 (2017), 853-894.  doi: 10.1007/s00161-017-0564-z.

[18]

M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptot. Anal., 112 (2019), 185-212.  doi: 10.3233/ASY-181502.

[19]

M. HeidaS. Neukamm and M. Varga, Stochastic homogenization of $\Lambda$-convex gradient flows, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 427-453.  doi: 10.3934/dcdss.2020328.

[20]

H. Hoppe, Homogenization of Rapidly Oscillating Riemannian Manifolds, Dissertation, TU Dresden, 2020, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-743766.

[21]

H. Hoppe, S. Neukamm and M. Schäffner, Stochastic homogenization of non-convex integral functionals with degenerate growth, (in preparation), 2021.

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[23]

S. M. Kozlov, Averaging of random operators, Mat. Sb., 109 (1979), 188-202. 

[24]

M. Liero and S. Reichelt, Homogenization of Cahn–Hilliard-type equations via evolutionary $\Gamma$-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.

[25]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. 

[26]

A. MielkeS. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353.

[27]

A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790.

[28]

S. Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Technische Universität München, 2010.

[29]

S. NeukammM. Schäffner and A. Schlömerkemper, Stochastic homogenization of nonconvex discrete energies with degenerate growth, SIAM J. Math. Anal., 49 (2017), 1761-1809.  doi: 10.1137/16M1097705.

[30]

S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230.

[31]

S. NeukammM. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, Asymptot. Anal., 125 (2021), 247-287. 

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.

[33]

G. C. Papanicolaou and S. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, 1 (1979), 835-873. 

[34]

M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342.

[35]

A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012.

[36]

C. Vogt, A homogenization theorem leading to a Volterra-integrodifferential equation for permeation chromotography, Preprint No 155, SFB 123, Heidelberg, 1982.

[37]

V. V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

[38]

V. V. Zhikov and A. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

K. T. Andrews and S. Wright, Stochastic homogenization of elliptic boundary-value problems with $L^p$-data, Asymptot. Anal., 17 (1998), 165-184. 

[3]

T. ArbogastJ. Douglas, Jr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[4]

E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), 570-598.  doi: 10.1137/0322035.

[5]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flow, Comptes Rendusa l'Académie des Sciences, 320 (1994), 1289–1294.

[6]

A. BourgeatA. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. 

[7]

D. CioranescuA. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math., 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028.

[8]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[9]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math., 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[10]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization., Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[12]

D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics. Springer-Verlag, New York, 1988.

[13]

J. Fischer and S. Neukamm, Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Arch. Ration. Mech. Anal., 242 (2021), 343-452.  doi: 10.1007/s00205-021-01686-9.

[14]

G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. 

[15]

H. Hanke, Homogenization in gradient plasticity, Math. Models Methods Appl. Sci., 21 (2011), 1651-1684.  doi: 10.1142/S0218202511005520.

[16]

M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022.

[17]

M. Heida, Stochastic homogenization of rate-independent systems and applications, Contin. Mech. Thermodyn., 29 (2017), 853-894.  doi: 10.1007/s00161-017-0564-z.

[18]

M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptot. Anal., 112 (2019), 185-212.  doi: 10.3233/ASY-181502.

[19]

M. HeidaS. Neukamm and M. Varga, Stochastic homogenization of $\Lambda$-convex gradient flows, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 427-453.  doi: 10.3934/dcdss.2020328.

[20]

H. Hoppe, Homogenization of Rapidly Oscillating Riemannian Manifolds, Dissertation, TU Dresden, 2020, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-743766.

[21]

H. Hoppe, S. Neukamm and M. Schäffner, Stochastic homogenization of non-convex integral functionals with degenerate growth, (in preparation), 2021.

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[23]

S. M. Kozlov, Averaging of random operators, Mat. Sb., 109 (1979), 188-202. 

[24]

M. Liero and S. Reichelt, Homogenization of Cahn–Hilliard-type equations via evolutionary $\Gamma$-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.

[25]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. 

[26]

A. MielkeS. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353.

[27]

A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790.

[28]

S. Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Technische Universität München, 2010.

[29]

S. NeukammM. Schäffner and A. Schlömerkemper, Stochastic homogenization of nonconvex discrete energies with degenerate growth, SIAM J. Math. Anal., 49 (2017), 1761-1809.  doi: 10.1137/16M1097705.

[30]

S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230.

[31]

S. NeukammM. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, Asymptot. Anal., 125 (2021), 247-287. 

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.

[33]

G. C. Papanicolaou and S. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, 1 (1979), 835-873. 

[34]

M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342.

[35]

A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012.

[36]

C. Vogt, A homogenization theorem leading to a Volterra-integrodifferential equation for permeation chromotography, Preprint No 155, SFB 123, Heidelberg, 1982.

[37]

V. V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

[38]

V. V. Zhikov and A. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.

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