Random Dirichlet problem: Scalar Darcy's law
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 \(L^ 2 (\Omega)\).
scientific article; zbMATH DE number 739488