January  2021, 14(1): 427-453. doi: 10.3934/dcdss.2020328

Stochastic homogenization of $ \Lambda $-convex gradient flows

1. 

Fakultät für Mathematik, Technische Universität München, Garching, Germany

2. 

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

3. 

Faculty of Mathematics, Technische Universität Dresden, Dresden, Germany

* Corresponding author: Mario Varga

This paper is dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  May 2019 Revised  October 2019 Published  April 2020

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $ \Lambda $-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $ p $-Laplace operator with $ p\in (1, \infty) $. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($ \Lambda $-)convex functionals.

Citation: Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Studies in Economic Theory, 4. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-03004-2.  Google Scholar

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

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.  Google Scholar

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

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H. Attouch, Convergence de fonctionnelles convexes, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 1–40.  Google Scholar

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H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

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V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

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V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

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

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H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. Google Scholar

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H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

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

[13]

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

[14]

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

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F. Delarue and R. Rhodes, Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.  doi: 10.3233/ASY-2008-0925.  Google Scholar

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Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260.   Google Scholar

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L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010. doi: 10.1090/gsm/019.  Google Scholar

[19]

A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.  doi: 10.1214/EJP.v13-591.  Google Scholar

[20]

T. FatimaA. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.  doi: 10.1080/00036811.2011.625016.  Google Scholar

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

[22]

H. Hanke and D. Knees, A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.  doi: 10.3233/ASY-161396.  Google Scholar

[23]

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.  Google Scholar

[24]

M. Heida, Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, 91 (2012), 1243-1264.  doi: 10.1080/00036811.2011.567191.  Google Scholar

[25]

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

[26]

A. Y. Kruger, On Fréchet subdifferentials, J. Math. Sci. (N.Y.), 116 (2003), 3325-3358.  doi: 10.1023/A:1023673105317.  Google Scholar

[27]

M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary $\Gamma$-convergence, Nonlinear Differ. Equat. Appl., 25 (2018), Art. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.  Google Scholar

[28]

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

[29]

P. Mathieu and A. Piatnitski, Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287-2307.  doi: 10.1098/rspa.2007.1876.  Google Scholar

[30]

A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.  doi: 10.3934/dcds.2015.35.2679.  Google Scholar

[31]

A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.  doi: 10.1007/978-3-319-26883-5_3.  Google Scholar

[32]

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.  Google Scholar

[33]

A. MielkeR. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), 253-310.  doi: 10.1007/s00526-011-0482-z.  Google Scholar

[34]

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.  Google Scholar

[35]

S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. Google Scholar

[36]

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

[37]

S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. Google Scholar

[38]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[39]

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.  Google Scholar

[40]

G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873.   Google Scholar

[41]

R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236.  Google Scholar

[42]

R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013.  Google Scholar

[43]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. Google Scholar

[44]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

[45]

M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409. doi: 10.4310/DPDE.2011.v8.n4.a1.  Google Scholar

[46]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.  doi: 10.3934/dcds.2011.31.1427.  Google Scholar

[47]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997.  Google Scholar

[48]

U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574.  Google Scholar

[49]

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

[50]

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

[51]

J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.  doi: 10.1007/s10231-009-0112-y.  Google Scholar

[52]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.  Google Scholar

[53]

V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236.  Google Scholar

[54]

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

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Studies in Economic Theory, 4. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-03004-2.  Google Scholar

[2]

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

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.  Google Scholar

[4]

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

[5]

H. Attouch, Convergence de fonctionnelles convexes, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 1–40.  Google Scholar

[6]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

[7]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[8]

V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[9]

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

[10]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. Google Scholar

[11]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal transportation, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 413 (2014), 100–144. arXiv: 1009.3737.  Google Scholar

[16]

F. Delarue and R. Rhodes, Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.  doi: 10.3233/ASY-2008-0925.  Google Scholar

[17]

Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260.   Google Scholar

[18]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010. doi: 10.1090/gsm/019.  Google Scholar

[19]

A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.  doi: 10.1214/EJP.v13-591.  Google Scholar

[20]

T. FatimaA. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.  doi: 10.1080/00036811.2011.625016.  Google Scholar

[21]

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

[22]

H. Hanke and D. Knees, A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.  doi: 10.3233/ASY-161396.  Google Scholar

[23]

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.  Google Scholar

[24]

M. Heida, Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, 91 (2012), 1243-1264.  doi: 10.1080/00036811.2011.567191.  Google Scholar

[25]

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

[26]

A. Y. Kruger, On Fréchet subdifferentials, J. Math. Sci. (N.Y.), 116 (2003), 3325-3358.  doi: 10.1023/A:1023673105317.  Google Scholar

[27]

M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary $\Gamma$-convergence, Nonlinear Differ. Equat. Appl., 25 (2018), Art. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.  Google Scholar

[28]

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

[29]

P. Mathieu and A. Piatnitski, Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287-2307.  doi: 10.1098/rspa.2007.1876.  Google Scholar

[30]

A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.  doi: 10.3934/dcds.2015.35.2679.  Google Scholar

[31]

A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.  doi: 10.1007/978-3-319-26883-5_3.  Google Scholar

[32]

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.  Google Scholar

[33]

A. MielkeR. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), 253-310.  doi: 10.1007/s00526-011-0482-z.  Google Scholar

[34]

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.  Google Scholar

[35]

S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. Google Scholar

[36]

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

[37]

S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. Google Scholar

[38]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[39]

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.  Google Scholar

[40]

G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873.   Google Scholar

[41]

R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236.  Google Scholar

[42]

R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013.  Google Scholar

[43]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. Google Scholar

[44]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.  Google Scholar

[45]

M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409. doi: 10.4310/DPDE.2011.v8.n4.a1.  Google Scholar

[46]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.  doi: 10.3934/dcds.2011.31.1427.  Google Scholar

[47]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997.  Google Scholar

[48]

U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.  doi: 10.1137/070684574.  Google Scholar

[49]

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

[50]

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

[51]

J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.  doi: 10.1007/s10231-009-0112-y.  Google Scholar

[52]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.  Google Scholar

[53]

V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236.  Google Scholar

[54]

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

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