American Institute of Mathematical Sciences

Advanced
November  2010, 9(6): 1753-1789. doi: 10.3934/cpaa.2010.9.1753

$\sum $-convergence and reiterated homogenization of nonlinear parabolic operators

Jean Louis Woukeng 1,

1. 

Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67 Dschang, Cameroon

Received  April 2009 Revised  September 2009 Published  August 2010

We study the reiterated homogenization of nonlinear parabolic differential equations associated with monotone operators. Contrary to what is usually done in the deterministic homogenization theory, we present a new approach based on a deterministic assumption on the coefficients of the operators, which allows us to consider the concrete homogenization problems from a true and natural perspective, taking into account the discontinuities in general. Based on this new approach we obtain very general homogenization results, and we solve several concrete homogenization problems. Our main tool is the theory of homogenization structures, and our homogenization approach falls within the scope of multiscale convergence method.
Keywords: nonlinear, Reiterated, homogenization structures, parabolic..
Mathematics Subject Classification: Primary: 35B40, 46J1.
Citation: Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[1]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[2]

Andrea Braides, Valeria Chiadò Piat. Non convex homogenization problems for singular structures. Networks & Heterogeneous Media, 2008, 3 (3) : 489-508. doi: 10.3934/nhm.2008.3.489
[3]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295
[4]

Mohamed Camar-Eddine, Laurent Pater. Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures. Networks & Heterogeneous Media, 2013, 8 (4) : 913-941. doi: 10.3934/nhm.2013.8.913
[5]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[6]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[7]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[8]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[9]

Aníbal Rodríguez-Bernal, Robert Willie. Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 385-410. doi: 10.3934/dcdsb.2005.5.385
[10]

E. N. Dancer. Some examples on solution structures for weakly nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 817-826. doi: 10.3934/dcds.2005.12.817
[11]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[12]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[13]

Frédéric Legoll, William Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 1-27. doi: 10.3934/dcdss.2015.8.1
[14]

M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721
[15]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[16]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612
[17]

Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451
[18]

Rita Ferreira, Elvira Zappale. Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1747-1793. doi: 10.3934/cpaa.2020072
[19]

Partha Guha, Indranil Mukherjee. Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1677-1695. doi: 10.3934/dcdsb.2018287
[20]

Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences