Homogenisation theory for Friedrichs systems

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May 2014, 13(3): 1017-1044. doi: 10.3934/cpaa.2014.13.1017

Homogenisation theory for Friedrichs systems

Krešimir Burazin ^1, and Marko Vrdoljak ^2,

1.	Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 Osijek, Croatia
2.	Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10 Zagreb, Croatia

Received March 2013 Revised September 2013 Published December 2013

Abstract
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We develop a general homogenisation procedure for Friedrichs systems. Under reasonable assumptions, the concepts of $G$ and $H$-convergence are introduced. As Friedrichs systems can be used to represent various boundary or initial-boundary value problems for partial differential equations, some additional assumptions are needed for compactness results. These assumptions are particularly examined for the stationary diffusion equation, the heat equation and a model example of a first order equation leading to memory effects. In the first two cases, the equivalence with the original notion of $H$-convergence is proved.

Keywords: compensated compactness., symmetric positive systems, H-convergence, Homogenisation, G-convergence.

Mathematics Subject Classification: Primary: 35B27, 35F45, 35M32; Secondary: 47F0.

Citation: 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

References:

[1]	Grégoire Allaire, Shape Optimization by the Homogenization Method,, Springer-Verlag, (2002).
[2]	Nenad Antonić and Krešimir Burazin, Graph spaces of first-order linear partial differential operators,, \emph{Math. Communications}, 14 (2009), 135.
[3]	Nenad Antonić and Krešimir Burazin, Intrinsic boundary conditions for Friedrichs systems,, \emph{Comm. Partial Diff. Eq.}, 35 (2010), 1690.
[4]	Nenad Antonić and Krešimir Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems,, \emph{J. Diff. Eq.}, 250 (2011), 3630.
[5]	Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Second-order equations as Friedrichs systems,, \emph{Nonlinear Analysis: RWA}, 14 (2014), 290.
[6]	Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Heat equation as a Friedrichs system,, \emph{Journal of Mathematical Analysis and Applications}, 404 (2013), 537.
[7]	Nenad Antonić and Marko Vrdoljak, Parabolic H-convergence and small-amplitude homogenization,, \emph{Applicable Analysis}, 88 (2009), 1493.
[8]	Alain Bensoussan, Jacques-Lous Lions and Georgios Papanicolaou, Asymptotic Analysis in Periodic Structures,, North\'c湌olland, (1978).
[9]	Krešimir Burazin, Contributions to the theory of Friedrichs' and Hyperbolic Systems,, Ph.D thesis (in Croatian), (2008).
[10]	Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas, A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs' systems,, \emph{SIAM J. Numer. Anal.}, 51 (2013), 1933.
[11]	Andrea Dall'Aglio and Francois Murat, A corrector result for $H$-converging parabolic problems with time-dependent coefficients,, \emph{Annali della Scuola Normale Superiore di Pisa}, 25 (1997), 329.
[12]	Daniele Antonio Di Pietro and Alexandre Ern, Mathematical Aspects of Discontinuous Galer\-Kin Methods,, Springer, (2012).
[13]	Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology, Vol. II,, Springer, (1992).
[14]	Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements,, Springer, (2004).
[15]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 753.
[16]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 2363.
[17]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. III. Multifield theories with partial coercivity,, \emph{SIAM J. Numer. Anal.}, 46 (2008), 776.
[18]	Alexandre Ern, Jean-Luc Guermond and Gilbert Caplain, An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems,, \emph{Comm. Partial Diff. Eq.}, 32 (2007), 317.
[19]	Kurt O. Friedrichs, Symmetric positive linear differential equations,, \emph{Comm. Pure Appl. Math.}, 11 (1958), 333.
[20]	Max Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions,, Ph.D thesis, (2004).
[21]	Jacques-Lous Lions and Enrico Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1,, Springer, (1972).
[22]	François Murat and Luc Tartar, H-convergence,, in \emph{S\'eminaire d'Analyse Fonctionnelle et Num\'erique de l'Universit\'e d'Alger, (1978), 21.
[23]	Jeffrey Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, \emph{Trans. Amer. Math. Soc.}, 291 (1985), 167.
[24]	Sergio Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore,, \emph{Ann. Scuola Norm. Sup. Pisa}, 21 (1967), 657.
[25]	Sergio Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1968), 571.
[26]	Sergio Spagnolo, Convergence of parabolic equations,, \emph{Bolletino della Unione Matematica Italiana}, 14 (1977), 547.
[27]	Luc Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles,, in \emph{Control Theory, 107 (1975), 420.
[28]	Luc Tartar, Nonlocal effects induced by homogenisation,, in \emph{Partial Differential Equations and the Calculus of Variation, (1989), 925.
[29]	Luc Tartar, The General Theory of Homogenization: A Personalized Introduction,, Springer, (2009).
[30]	Marcus Waurick, On the theory of homogenization of evolutionary equations in Hilbert spaces,, preprint, ().
[31]	Vasili V. Žikov, Sergei M. Kozlov, Ol'ga A. Oleinik and Ha Tien Ngoan, Homogenization and G-convergence of differential operators,, \emph{Uspehi Mat. Nauk}, 34 (1979), 65.
[32]	Vasili V. Žikov, Sergei M. Kozlov and Ol'ga A. Oleinik, On G-convergence of parabolic operators,, \emph{Russ. Math. Surv.}, 36 (1981), 9.

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References:

[1]	Grégoire Allaire, Shape Optimization by the Homogenization Method,, Springer-Verlag, (2002).
[2]	Nenad Antonić and Krešimir Burazin, Graph spaces of first-order linear partial differential operators,, \emph{Math. Communications}, 14 (2009), 135.
[3]	Nenad Antonić and Krešimir Burazin, Intrinsic boundary conditions for Friedrichs systems,, \emph{Comm. Partial Diff. Eq.}, 35 (2010), 1690.
[4]	Nenad Antonić and Krešimir Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems,, \emph{J. Diff. Eq.}, 250 (2011), 3630.
[5]	Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Second-order equations as Friedrichs systems,, \emph{Nonlinear Analysis: RWA}, 14 (2014), 290.
[6]	Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Heat equation as a Friedrichs system,, \emph{Journal of Mathematical Analysis and Applications}, 404 (2013), 537.
[7]	Nenad Antonić and Marko Vrdoljak, Parabolic H-convergence and small-amplitude homogenization,, \emph{Applicable Analysis}, 88 (2009), 1493.
[8]	Alain Bensoussan, Jacques-Lous Lions and Georgios Papanicolaou, Asymptotic Analysis in Periodic Structures,, North\'c湌olland, (1978).
[9]	Krešimir Burazin, Contributions to the theory of Friedrichs' and Hyperbolic Systems,, Ph.D thesis (in Croatian), (2008).
[10]	Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas, A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs' systems,, \emph{SIAM J. Numer. Anal.}, 51 (2013), 1933.
[11]	Andrea Dall'Aglio and Francois Murat, A corrector result for $H$-converging parabolic problems with time-dependent coefficients,, \emph{Annali della Scuola Normale Superiore di Pisa}, 25 (1997), 329.
[12]	Daniele Antonio Di Pietro and Alexandre Ern, Mathematical Aspects of Discontinuous Galer\-Kin Methods,, Springer, (2012).
[13]	Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology, Vol. II,, Springer, (1992).
[14]	Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements,, Springer, (2004).
[15]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 753.
[16]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 2363.
[17]	Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. III. Multifield theories with partial coercivity,, \emph{SIAM J. Numer. Anal.}, 46 (2008), 776.
[18]	Alexandre Ern, Jean-Luc Guermond and Gilbert Caplain, An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems,, \emph{Comm. Partial Diff. Eq.}, 32 (2007), 317.
[19]	Kurt O. Friedrichs, Symmetric positive linear differential equations,, \emph{Comm. Pure Appl. Math.}, 11 (1958), 333.
[20]	Max Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions,, Ph.D thesis, (2004).
[21]	Jacques-Lous Lions and Enrico Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1,, Springer, (1972).
[22]	François Murat and Luc Tartar, H-convergence,, in \emph{S\'eminaire d'Analyse Fonctionnelle et Num\'erique de l'Universit\'e d'Alger, (1978), 21.
[23]	Jeffrey Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, \emph{Trans. Amer. Math. Soc.}, 291 (1985), 167.
[24]	Sergio Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore,, \emph{Ann. Scuola Norm. Sup. Pisa}, 21 (1967), 657.
[25]	Sergio Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1968), 571.
[26]	Sergio Spagnolo, Convergence of parabolic equations,, \emph{Bolletino della Unione Matematica Italiana}, 14 (1977), 547.
[27]	Luc Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles,, in \emph{Control Theory, 107 (1975), 420.
[28]	Luc Tartar, Nonlocal effects induced by homogenisation,, in \emph{Partial Differential Equations and the Calculus of Variation, (1989), 925.
[29]	Luc Tartar, The General Theory of Homogenization: A Personalized Introduction,, Springer, (2009).
[30]	Marcus Waurick, On the theory of homogenization of evolutionary equations in Hilbert spaces,, preprint, ().
[31]	Vasili V. Žikov, Sergei M. Kozlov, Ol'ga A. Oleinik and Ha Tien Ngoan, Homogenization and G-convergence of differential operators,, \emph{Uspehi Mat. Nauk}, 34 (1979), 65.
[32]	Vasili V. Žikov, Sergei M. Kozlov and Ol'ga A. Oleinik, On G-convergence of parabolic operators,, \emph{Russ. Math. Surv.}, 36 (1981), 9.

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