The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary

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2013, 2013(special): 85-94. doi: 10.3934/proc.2013.2013.85

The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary

Carmen Calvo-Jurado ^1, , Juan Casado-Díaz ^2, and Manuel Luna-Laynez ^2,

1.	Dpto. de Matemáticas. Escuela Politécnica, Avenida de la Universidad s/n, 10003 Cáceres, Spain
2.	Dpto. de Ecuaciones Diferenciales y Análisis Numérico., Fac. de Matemáticas. C. Tarfia s/n., 41012 Sevilla

Received September 2012 Published November 2013

Abstract
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We consider a domain in $\mathbb{R}^N$, $N\geq 3$, such that a portion of its boundary is plane. In this portion we fix a sequence $K_\epsilon$ of small subsets randomly distributed in such way that the distance between them is of order $\epsilon$ and their diameters are of order $\epsilon^\frac{N-1}{N-2}$. We study the asymptotic behavior of the heat equation with Dirichlet conditions on $K_\epsilon$ and Neumann conditions on the rest of the boundary. We prove the convergence to a limit problem with a Fourier-Robin boundary condition which has the physical interest of being deterministic.

Keywords: Homogenization, random perforated domains., linear problems.

Mathematics Subject Classification: Primary: 35B27; Secondary: 35R6.

Citation: Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary. Conference Publications, 2013, 2013 (special) : 85-94. doi: 10.3934/proc.2013.2013.85

References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math Anal. 23 (1992), 23 (1992), 1482. Google Scholar
[2]	G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation,, Proc. Roy. Soc. Edinburgh A 456* (1996), 456* (1996), 297. Google Scholar
[3]	T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory,, SIAM J. Math. Anal., 21 (1990), 823. Google Scholar
[4]	A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math. 456* (1994) 19-51., 456* (1994), 19. Google Scholar
[5]	L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem,, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 26 (2009), 375. Google Scholar
[6]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets,, Disc. Cont. Din. Systems Suplement (2007), (2007), 181. Google Scholar
[7]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets., ESAIM Control, 15 (2009), 49. Google Scholar
[8]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Homogenization of Dirichlet problems in randomly perforated domains., To appear., (). Google Scholar
[9]	J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains,, Proc. Royal Soc. Edinburgh, 127A (1997), 457. Google Scholar
[10]	J. Casado-Díaz, Two-Scale convergence for nonlinear Dirichlet problems in perforated domains,, Proceedings of the Royal Society of Edinburgh A, 130 (2000), 249. Google Scholar
[11]	J. Casado-Díaz, I. Gayte, The Two-Scale Convergence Method Applied to Generalized Besicovitch Spaces,, Proc. R. Soc. Lond. A 2002 458, 458 (2002), 2925. Google Scholar
[12]	J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains,, SIAM J. Math. Anal., 31 (2000), 581. Google Scholar
[13]	D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization,, C.R. Acad. Sci. Paris, 335 (2002), 99. Google Scholar
[14]	D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, in, Nonlinear Partial Differential Equations and Their Applications, II (1982), 98. Google Scholar
[15]	G. Dal Maso, L. Modica, Nonlinear stochastic homogenization,, Ann. Mat. Pura Appl. 72 (1993), 72 (1993), 405. Google Scholar
[16]	G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory,, J. Reine Angew, 368 (1986), 28. Google Scholar
[17]	G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence,, Appl. Math. Optim. 15 (1987), 15 (1987), 15. Google Scholar
[18]	G. Dal Maso, F. Murat, Asymptotic behaviour and correctors for the Dirichlet problems in perforated domains with homogeneous monotone operators,, Ann. Sc. Norm. Sup. Pisa. 7, 4 (1997), 765. Google Scholar
[19]	V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals,, Springer-Verlag, (1994). Google Scholar
[20]	S.M. Kozlov, Homogenization of random operators,, Math. U.S.S.R. Sb., 37 (1980), 167. Google Scholar
[21]	M. Lenczner, Homogénéisation d'un circuit électrique,, C. R. Acad. Sci. Paris II, 324 (1997), 537. Google Scholar
[22]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal. 20 (1989), 20 (1989), 608. Google Scholar
[23]	G.C. Papanicolaou, S.R.S. Varadhan, Diffusion in regions with many small holes,, in Stochastic differential systems, (1980). Google Scholar
[24]	G.C. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, Colloq. Math. Soc. J. Bolyai, (1981), 835. Google Scholar
[25]	I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure,, Mat. Sb. 187, 8 (1996), 125. Google Scholar
[26]	Yosida, K., Functional Analysis,, Springer-Verlag, (1980). Google Scholar
[27]	V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients,, Sib. Math. J. 21, 21 (1981), 470. Google Scholar

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References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math Anal. 23 (1992), 23 (1992), 1482. Google Scholar
[2]	G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation,, Proc. Roy. Soc. Edinburgh A 456* (1996), 456* (1996), 297. Google Scholar
[3]	T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory,, SIAM J. Math. Anal., 21 (1990), 823. Google Scholar
[4]	A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math. 456* (1994) 19-51., 456* (1994), 19. Google Scholar
[5]	L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem,, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 26 (2009), 375. Google Scholar
[6]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets,, Disc. Cont. Din. Systems Suplement (2007), (2007), 181. Google Scholar
[7]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets., ESAIM Control, 15 (2009), 49. Google Scholar
[8]	C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Homogenization of Dirichlet problems in randomly perforated domains., To appear., (). Google Scholar
[9]	J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains,, Proc. Royal Soc. Edinburgh, 127A (1997), 457. Google Scholar
[10]	J. Casado-Díaz, Two-Scale convergence for nonlinear Dirichlet problems in perforated domains,, Proceedings of the Royal Society of Edinburgh A, 130 (2000), 249. Google Scholar
[11]	J. Casado-Díaz, I. Gayte, The Two-Scale Convergence Method Applied to Generalized Besicovitch Spaces,, Proc. R. Soc. Lond. A 2002 458, 458 (2002), 2925. Google Scholar
[12]	J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains,, SIAM J. Math. Anal., 31 (2000), 581. Google Scholar
[13]	D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization,, C.R. Acad. Sci. Paris, 335 (2002), 99. Google Scholar
[14]	D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, in, Nonlinear Partial Differential Equations and Their Applications, II (1982), 98. Google Scholar
[15]	G. Dal Maso, L. Modica, Nonlinear stochastic homogenization,, Ann. Mat. Pura Appl. 72 (1993), 72 (1993), 405. Google Scholar
[16]	G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory,, J. Reine Angew, 368 (1986), 28. Google Scholar
[17]	G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence,, Appl. Math. Optim. 15 (1987), 15 (1987), 15. Google Scholar
[18]	G. Dal Maso, F. Murat, Asymptotic behaviour and correctors for the Dirichlet problems in perforated domains with homogeneous monotone operators,, Ann. Sc. Norm. Sup. Pisa. 7, 4 (1997), 765. Google Scholar
[19]	V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals,, Springer-Verlag, (1994). Google Scholar
[20]	S.M. Kozlov, Homogenization of random operators,, Math. U.S.S.R. Sb., 37 (1980), 167. Google Scholar
[21]	M. Lenczner, Homogénéisation d'un circuit électrique,, C. R. Acad. Sci. Paris II, 324 (1997), 537. Google Scholar
[22]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal. 20 (1989), 20 (1989), 608. Google Scholar
[23]	G.C. Papanicolaou, S.R.S. Varadhan, Diffusion in regions with many small holes,, in Stochastic differential systems, (1980). Google Scholar
[24]	G.C. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, Colloq. Math. Soc. J. Bolyai, (1981), 835. Google Scholar
[25]	I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure,, Mat. Sb. 187, 8 (1996), 125. Google Scholar
[26]	Yosida, K., Functional Analysis,, Springer-Verlag, (1980). Google Scholar
[27]	V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients,, Sib. Math. J. 21, 21 (1981), 470. Google Scholar

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