# American Institute of Mathematical Sciences

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. [2] G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh A 456 (1996), 297-342. [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. [4] A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994) 19-51. [5] L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 2, 375-395. [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. [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. [8] C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Homogenization of Dirichlet problems in randomly perforated domains., To appear., (). [9] J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains, Proc. Royal Soc. Edinburgh, 127A (1997), 457-478. [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. [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. [12] J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal., 31 (2000), 581-624. [13] D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, I, 335 (2002), 99-104. [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. [15] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 72 (1993), 405-414. [16] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew, Math. 368 (1986), 28-42. [17] G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence, Appl. Math. Optim. 15 (1987), 15-63. [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. [19] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. [20] S.M. Kozlov, Homogenization of random operators, Math. U.S.S.R. Sb., 37 (1980), 167-180. [21] M. Lenczner, Homogénéisation d'un circuit électrique, C. R. Acad. Sci. Paris II, 324 (1997), 537-542. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608-623. [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. [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. [25] I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure, Mat. Sb. 187, 8 (1996), 125-157. [26] Yosida, K., Functional Analysis, Springer-Verlag, Berlin 1980. [27] V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients, Sib. Math. J. 21, (3) 470-482 (1981).

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math Anal. 23 (1992),1482-1512. [2] G. Allaire, M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh A 456 (1996), 297-342. [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. [4] A. Bourgeat, A. Mikelic, S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994) 19-51. [5] L.A. Caffarelli, A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linèaire 26 (2009), 2, 375-395. [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. [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. [8] C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez, Homogenization of Dirichlet problems in randomly perforated domains., To appear., (). [9] J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains, Proc. Royal Soc. Edinburgh, 127A (1997), 457-478. [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. [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. [12] J. Casado-Díaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal., 31 (2000), 581-624. [13] D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, I, 335 (2002), 99-104. [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. [15] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 72 (1993), 405-414. [16] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew, Math. 368 (1986), 28-42. [17] G. Dal Maso, U. Mosco, Wiener-criterion and $\Gamma$-convergence, Appl. Math. Optim. 15 (1987), 15-63. [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. [19] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. [20] S.M. Kozlov, Homogenization of random operators, Math. U.S.S.R. Sb., 37 (1980), 167-180. [21] M. Lenczner, Homogénéisation d'un circuit électrique, C. R. Acad. Sci. Paris II, 324 (1997), 537-542. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608-623. [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. [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. [25] I.V. Skrypnik., Averaging of nonlinear Dirichlet problems in punctured domains of general structure, Mat. Sb. 187, 8 (1996), 125-157. [26] Yosida, K., Functional Analysis, Springer-Verlag, Berlin 1980. [27] V.V. Yurinski, Averaging an elliptic boundary value problem with random coefficients, Sib. Math. J. 21, (3) 470-482 (1981).
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