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A one-dimensional case of stochastic homogenization. (Italian. English summary) Zbl 0534.49011

The aim of the paper is to prove that the functionals \[ F_ n(u)=\int^{1}_{0}a_ x(t)| u'(t)|^ 2dt \] \(\Gamma\)- converge stochastically to the functional \[ F(u)=\frac{2\lambda \Lambda}{\lambda +\Lambda}\int^{1}_{0}| u'(t)|^ 2dt,\quad \lambda,\quad \Lambda \in {\mathbb{R}}^+,\quad n\in {\mathbb{N}}. \] In order to explain what stochastic \(\Gamma\)-convergence is, consider the space \({\mathfrak F}\) of functionals \[ F_ a(u)=\int^{1}_{0}a(t)| u'(t)|^ 2dt \] with \(a\in L^{\infty}(0,1)\) and \(\lambda \leq a(t)\leq \Lambda,\) denote by \(L_ n\) the uniform probability on \(Z_ n\), and define \(f_ n:Z_ n\to {\mathfrak F}\) by setting \(f_ n(x)=F_{a_ x}.\) Then, the stochastic \(\Gamma\)-convergence of \(F_ n\) to F is the weak convergence (with respect to a suitable topology on \({\mathfrak F})\) of the sequence of measures \(f_ n(L_ n)\) to the Dirac measure \(\delta_{F_{\infty}}\).
Reviewer: G.Buttazzo

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
60F17 Functional limit theorems; invariance principles
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