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), 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

show all references

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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