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Homogenization of random semilinear PDEs. (English) Zbl 0989.35022
A homogenization problem for a system of semilinear random parabolic equations \[ \partial_{t}u^\varepsilon + \frac 12 e^{2V(x/\varepsilon,\omega)} \text{div}\left(a\Bigl({x\over\varepsilon},\omega\Bigr)e^{-2 V(x/\varepsilon,\omega)}\nabla u^\varepsilon\right) + h(x, u^\varepsilon,\nabla u^\varepsilon) = 0, \;u^\varepsilon(T, \cdot) = H, \] with \(u=(u_1,\ldots,u_{m})\), \(x\in\mathbb R^{d}\), and \(t\in [0,T]\), is concerned. Let \((\Omega,\mu)\) be a probability space, \((\tau_{x}, x\in\mathbb R^{d})\) be a group of transformations of \(\Omega\) satisfying measurability and stochastic continuity assumptions and such that \(\mu\) is invariant and ergodic under the action of \((\tau_{x})\). Let \(\mathbf V:\Omega\to\mathbb R\) be an essentially bounded measurable function, \(\mathbf a:\Omega\to\mathbb R^{d}\otimes \mathbb R^{d}\) a measurable function whose values are symmetric matrices; the random fields \(a\), \(V\) are defined by \(a(x,\omega) = \mathbf a(\tau _{x}\omega)\) and \(V(x,\omega) = \mathbf V(\tau_{x}\omega)\). Assume that \(\mathbf a\) is positive definite and bounded \(\mu\)-almost surely and \(\langle \mu, e^{-2\mathbf V}\rangle = 1\). Let the function \(H:\mathbb R^{d}\to\mathbb R^{m}\) be continuous and square integrable, let \(h:\mathbb R^{d}\times\mathbb R^{m}\times\mathbb R^{m \times d}\to\mathbb R^{m}\) be a bounded uniformly continuous function satisfying some Lipschitzianity and monotonicity hypotheses. Then \(\lim_{\varepsilon\to 0}\int_\Omega\int_{G}\|u^\varepsilon(t,x)-u^0(t,x)\|^{p} dx d\mu =0\) for all bounded domains \(G\subset \mathbb R^{d}\), all \(t\in[0,T]\) and all \(p\in\mathopen]1,Q\mathclose[\) with certain \(Q\in\mathopen ]1,2\mathclose]\). Here \(u^0\) denotes the solution to the homogenized system \(\partial_{t}u^0 + \frac 12\overline a_{ij}\partial^2_{ij} u^0 + \overline h(x,u^0,\nabla u^0)=0\), \(u^0(T,\cdot) = H\). The coefficients \(\overline a\), \(\overline h\) are defined by \(\overline a = \int_\Omega (\text{Id}+\phi)\mathbf a(\text{Id}+\phi)^* e^{-2\mathbf V} d\mu\), \(\overline h(x,y,z)= \int_\Omega h(x,y,z(\text{Id}+\phi))e^{ -2\mathbf V} d\mu\), where \(\phi = (\phi^{i}_{j})\) and \(\phi^{i}\) are vortex-free stationary fields solving a suitable auxiliary problem.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R60 PDEs with randomness, stochastic partial differential equations
60K37 Processes in random environments
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