## Random Dirichlet problem: Scalar Darcy’s law.(English)Zbl 0821.60029

Let $$T$$ be a finite family of compact sets of $$Y = (0,1)^ d$$ and $$Q$$ a probability measure on $$T$$. For $$z \in R^ d$$, put $$T_ z = \{K + z\}_{K \in T}$$. Denote by $$Q_ z$$ the corresponding transformed measure of $$Q$$ on $$T_ z$$. Define the Bernoulli space $$(\Sigma, {\mathcal P})$$ by $$(\Sigma, {\mathcal P}) = \prod_{z \in Z^ d} (T_ z, Q_ z)$$. For $$\omega \in (\omega_ z)_{z \in Z^ d} \in \Sigma$$, let $$K (\omega)$$ be the random holes in $$R^ d$$ given by $$K (\omega) = \bigcup_{z \in Z^ d} \omega_ z$$. Take a bounded domain $$\Omega$$ of $$R^ d$$. For $$\varepsilon_ n > 0$$, let $$u_ n = u_ n (\omega, \cdot)$$ be the solution of the random Dirichlet problem in $$\Omega \backslash K_ n (\omega) : - \Delta u_ n = f$$ in $$\Omega \backslash K_ n (\omega)$$, $$u_ n = 0$$ on $$\partial K_ n (\omega) \cup \partial \Omega$$, where $$K_ n (\omega) = \varepsilon_ n K (\omega)$$. Then, it is known that $$\lim_{n \to \infty} u_ n = 0$$ strongly in $$H^ 1_ 0 (\Omega)$$.
The main purpose of this paper is to find the rate of convergence of $$u_ n$$ to zero. For the solution $$v^ k (\omega, \cdot)$$ of the Dirichlet problem $$- \Delta v^ k(\omega, \cdot) = 1$$ in $$kY - K (\omega)$$, $$v^ k (\omega, \cdot) = 0$$ on $$K (\omega)$$ put $$\lambda_ k (\omega) = (1/k^ d) \int_{kY} v^ k (\omega, \lambda) dx$$. The authors use the ergodic theorem to show that $$\lambda_ k (\omega)$$ converges almost surely to $$\lambda = \sup_ k E (\lambda_ k (\omega))$$. Further, by using the epiconvergence method and its variational properties, the authors prove that $$u_ n (\omega, \cdot)/ \varepsilon^ 2_ n$$ converges weakly to $$\lambda f$$ in $$L^ 2 (\Omega)$$.

### MSC:

 60Fxx Limit theorems in probability theory 60J45 Probabilistic potential theory 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J25 Boundary value problems for second-order elliptic equations 60G10 Stationary stochastic processes 74E05 Inhomogeneity in solid mechanics
Full Text:

### References:

 [1] Ackoglu, M. A. and Krengel, U.: ?Ergodic Theorems for Superadditive Processes,?J. Reine. Angew. Math. 323 (1981), 53-67. · Zbl 0453.60039 [2] Allaire, G.: Homogénéisation des équations de Stokes et de Navier-Stokes, Thèse de Doctorat de l’Université Paris VI, Spécialité Mathématiques (1989). [3] Attouch, H.:Variational Convergence for Functions and Operators, Research Notes in Mathematics, Pitman, London. · Zbl 0561.49012 [4] Aubin, J. P. and Frankowska, H.:Set Valued Analysis, Birkhäuser (1990). [5] Brillard, A.: Homogénéisation de quelques équations de la mécanique des milieux continus, Thèse d’état (1990). [6] Castaing, C. and Valadier, M.:Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer Verlag (1977). · Zbl 0346.46038 [7] Chabi, E. and Michaille, G.: ?Ergodic Theory and Application to Non Convex Homogenization,?Set-Valued Analysis 2 (1994), 117-134. · Zbl 0813.28010 [8] Dal Maso, G. and Modica, L.: ?Nonlinear Stochastic Homogenization and Ergodic Theory,?J. Reine. Angew. Math. 368 (1986), 28-42. · Zbl 0582.60034 [9] Dal Maso, G. and Modica, L.: ?A general theory of variational functionals,? inTopics in Functional Analysis 1980-81, by F. Strocchi, E. Zarantonello, E. De Giorgi, G. Dal Maso, L. Modica, Scuola Superiore, Pisa, pp. 149-221. [10] De Giorgi, E.:Convergence Problems for Functions and Operators, Proc. Int. Meeting on ?Recent methods in nonlinear analysis?, Rene (1978), ed. E. De Giorgi, E. Magenes, Mosco Pitagora, Bologna, pp. 131-188. [11] Krengel, U.:Ergodic Theorems, Walter de Gruyter, Studies in Mathematics (1985). · Zbl 0575.28009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.