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

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

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.
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),1482-1512.  Google Scholar

[2]

G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh A 456 (1996), 297-342.  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), 823-836.  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.  Google Scholar

[5]

L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 2, 375-395.  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), 181-190.  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, Optim. and Calc. Var. 15 (2009), 49-67.  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), 457-478.  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) 249-276.  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, 2925-2946.  Google Scholar

[12]

J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal., 31 (2000), 581-624.  Google Scholar

[13]

D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, I, 335 (2002), 99-104.  Google Scholar

[14]

D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vols. II and III, Research Notes in Math. 60 and 70 Pitman, London, 1982, 98-138 and 154-178. Google Scholar

[15]

G. Dal Maso, L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 72 (1993), 405-414. Google Scholar

[16]

G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew, Math. 368 (1986), 28-42.  Google Scholar

[17]

G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence, Appl. Math. Optim. 15 (1987), 15-63.  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), 765-803.  Google Scholar

[19]

V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.  Google Scholar

[20]

S.M. Kozlov, Homogenization of random operators, Math. U.S.S.R. Sb., 37 (1980), 167-180. Google Scholar

[21]

M. Lenczner, Homogénéisation d'un circuit électrique, C. R. Acad. Sci. Paris II, 324 (1997), 537-542. Google Scholar

[22]

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

[23]

G.C. Papanicolaou, S.R.S. Varadhan, Diffusion in regions with many small holes, in Stochastic differential systems, Lecture Notes in Control and Information Sci., 25, Springer, Berlin, 1980, 190206.  Google Scholar

[24]

G.C. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Colloq. Math. Soc. J. Bolyai, North-Holland, Amsterdam (1981), 835-873.  Google Scholar

[25]

I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure, Mat. Sb. 187, 8 (1996), 125-157.  Google Scholar

[26]

Yosida, K., Functional Analysis, Springer-Verlag, Berlin 1980.  Google Scholar

[27]

V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients, Sib. Math. J. 21, (3) 470-482 (1981).  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math Anal. 23 (1992),1482-1512.  Google Scholar

[2]

G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh A 456 (1996), 297-342.  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), 823-836.  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.  Google Scholar

[5]

L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 2, 375-395.  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), 181-190.  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, Optim. and Calc. Var. 15 (2009), 49-67.  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), 457-478.  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) 249-276.  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, 2925-2946.  Google Scholar

[12]

J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal., 31 (2000), 581-624.  Google Scholar

[13]

D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, I, 335 (2002), 99-104.  Google Scholar

[14]

D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vols. II and III, Research Notes in Math. 60 and 70 Pitman, London, 1982, 98-138 and 154-178. Google Scholar

[15]

G. Dal Maso, L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 72 (1993), 405-414. Google Scholar

[16]

G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew, Math. 368 (1986), 28-42.  Google Scholar

[17]

G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence, Appl. Math. Optim. 15 (1987), 15-63.  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), 765-803.  Google Scholar

[19]

V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.  Google Scholar

[20]

S.M. Kozlov, Homogenization of random operators, Math. U.S.S.R. Sb., 37 (1980), 167-180. Google Scholar

[21]

M. Lenczner, Homogénéisation d'un circuit électrique, C. R. Acad. Sci. Paris II, 324 (1997), 537-542. Google Scholar

[22]

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

[23]

G.C. Papanicolaou, S.R.S. Varadhan, Diffusion in regions with many small holes, in Stochastic differential systems, Lecture Notes in Control and Information Sci., 25, Springer, Berlin, 1980, 190206.  Google Scholar

[24]

G.C. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Colloq. Math. Soc. J. Bolyai, North-Holland, Amsterdam (1981), 835-873.  Google Scholar

[25]

I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure, Mat. Sb. 187, 8 (1996), 125-157.  Google Scholar

[26]

Yosida, K., Functional Analysis, Springer-Verlag, Berlin 1980.  Google Scholar

[27]

V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients, Sib. Math. J. 21, (3) 470-482 (1981).  Google Scholar

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