Stochastic-periodic homogenization of Maxwell's equations with linear and periodic conductivity (Q520417)

The authors study the limiting behavior, as \(0<\varepsilon\to0\), of the sequence of solutions of the electromagnetic field governed by the following system: \(\partial_{t}D_{\varepsilon}(x,t,\omega)+J_{\varepsilon} (x,t,\omega)=\text{curl}\,H_{\varepsilon}(x,t,\omega)+F_{\varepsilon}(x,t,\omega)\) in \(Q\times \Lambda\), \(\partial_{t} B_{\varepsilon}(x,t,\omega)=-\text{curl}\, E_{\varepsilon}(x,t,\omega)\) in \(Q\times\Lambda\), \(\text{div} B_{\varepsilon}(x,t,\omega)=0\) in \(Q\times\Lambda\), \(\text{div} D_{\varepsilon}(x,t,\omega)\in L^2(Q\times\Lambda)\), \(\gamma\wedge E_{\varepsilon}(x,t,\omega)=0\in \mathbb{R}^3\) on \(\partial\Omega\times(0,T)\times\Lambda\), with initial conditions \(E_{\varepsilon}(\cdot,0,\cdot)=E_{\varepsilon}^0\) in \(\Omega\times\Lambda\), \(H_{\varepsilon}(\cdot,0,\cdot)=H_{\varepsilon}^0\) in \(\Omega\times\Lambda\), where \(B_{\varepsilon}^{i}(x,t,\omega)=\mu_{ij}(x,{\mathcal T}(x/\varepsilon)\omega,x/\varepsilon^2)H_{\varepsilon}^{j}(x,t,\omega)\), \(D_{\varepsilon}^{i}(x,t,\omega)=\eta_{ij}(x,{\mathcal T}(x/\varepsilon)\omega,x/\varepsilon^2)E_{\varepsilon}^{j}(x,t,\omega)\), \(J_{\varepsilon}^{i}(x,t,\omega)=\sigma_{ij}(x,{\mathcal T}(x/\varepsilon)\omega,x/\varepsilon^2)E_{\varepsilon}^{j}(x,t,\omega)\), \(i,j=1,2,3\), \(Q=\Omega\times(0,T)\), \(\Omega\) is a bounded open and simply connected set in \(\mathbb R^3\), \(\Lambda\) is a support of a probability space, \({\mathcal T}\) is a 3-dimensional dynamical system on \(\Lambda\), \(\gamma\) denotes the outer unit normal to \(\partial\Omega\) with respect to \(\Omega\). It is shown that the sequence of solutions to considered class of highly oscillatory problems converge to the solution of a homogenized Maxwell equation.