Existence and homogenization for a singular problem through rough surfaces (Q2836026)

Let \(\omega\) be a bounded smooth domain in \(\mathbb{R}^{N-1}\) (\(N\geq 2\)), let \(l>0\), and set \(Q:=\omega\times ]-l,l[\). Moreover, let \(g:]0,1[^{N-1}\rightarrow \mathcal{R}\) be a periodic positive Lipschitz continuous function and \(A:]0,1[^N\rightarrow \mathbb{R}^N\times \mathbb{R}^N\) be a \(]0,1[^N\)-periodic matrix field such that, for some \(\alpha,\beta>0\) and for all \(\lambda \in \mathbb{R}^N\), \((A(y)\lambda,\lambda)\geq \alpha |\lambda|^2\) and \(|A(y)\lambda|\leq \beta \lambda\), a.e. in \(]0,1[^N\). Finally, let \(h\) be a \(]0,1[^{N-1}\)-periodic function such that, for some \(h_0,M>0\), \(h_0<h(y)<M\) a.e. in \(\Gamma :=\{y=(y',y_N)\in ]0,1[^{N-1}\times \mathbb{R}: y_N=g(y')\}\), let \(\varepsilon,\kappa>0\), and set \(\Gamma_\varepsilon=\{(x',x_N)\in \omega \times \mathbb{R}: x_N=\varepsilon^\kappa g(\frac{x'}{\varepsilon})\}\), \(A^\varepsilon(x)=A(\frac{x}{\varepsilon})\), \(h^\varepsilon(x)=h(\frac{x}{\varepsilon})\).NEWLINENEWLINEThis paper deals with the following problem \((P_\varepsilon)\): NEWLINE\[NEWLINE-\operatorname{div}(A^\varepsilon\nabla u_\varepsilon)=f\zeta(u_\varepsilon)\quad\text{in }Q \setminus \Gamma_\varepsilon,NEWLINE\]NEWLINE NEWLINE\[NEWLINE(A^\varepsilon\nabla u_\varepsilon)_1 \nu_\varepsilon =(A^\varepsilon\nabla u_\varepsilon)_2 \nu_\varepsilon\quad\text{on }\Gamma_\varepsilon,NEWLINE\]NEWLINE NEWLINE\[NEWLINE(A^\varepsilon\nabla u_\varepsilon)_1 \nu_\varepsilon =-\varepsilon^\gamma h^\varepsilon (u_{\varepsilon 1}-u_{\varepsilon 2})\quad\text{on }\Gamma_\varepsilon,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_\varepsilon=0\quad\text{on }\partial Q,NEWLINE\]NEWLINE which is a model for the stationary heat diffusion in the medium \(Q\) made up by the two connected composite components \(Q_{\varepsilon 1}=\{(x',x_N)\in Q: x_N>\varepsilon^\kappa g(\frac{x'}{\varepsilon})\}\), \(Q_{\varepsilon 2}=\{(x',x_N)\in Q: x_N<\varepsilon^\kappa g(\frac{x'}{\varepsilon})\}\) separated by the interface \(\Gamma_\varepsilon\). The parameter \(\kappa\) describes, as \(\varepsilon\rightarrow 0\), the type of oscillation of the interface: for \(\kappa>1\) the interface tends to become flat, while for \(0<\kappa<1\) the interface becomes highly oscillating. In problem \((P_\varepsilon)\), the subscripts \(1,2\) denote the restriction, respectively, to \(Q_{\varepsilon 1}\) and \(Q_{\varepsilon 2}\), \(\nu_\varepsilon\) is the unit outward normal to \(Q_{\varepsilon 1}\), \(\gamma\) is a real number, \(\zeta\in C^0([0,+\infty)\) is a nonnegative function such that \(\zeta(s)\leq s^{-\theta}\) in \(]0,+\infty[\), with \(0<\theta\leq 1\), and \(f\in L^r(Q)\) for \(r>\frac{2}{1+\theta}\), with \(f\) non-zero and nonnegative in \(Q\).NEWLINENEWLINEUnder the above conditions, the authors prove the existence of at least a solution of problem \((P_\varepsilon)\) for each \(\varepsilon>0\). They also prove that any solution of \((P_\varepsilon)\) is bounded if \(f\in L^r(Q)\) for \(r>\frac{N}{2}\), and that problem \((P_\varepsilon)\) admits a unique solution if \(\zeta\) is nondecreasing.NEWLINENEWLINEHomogenization results are also established. In particular, the authors prove that there exists a subsequence of solutions \(\{u^{\varepsilon_k}\}\), \((\varepsilon_k\rightarrow 0)\) and a function \(u_0\) such that \(u^{\varepsilon_k}\rightarrow u_0\) strongly in \(L^2(Q)\), \(\chi_{Q_{\varepsilon_ki}}\nabla u^{\varepsilon_k}\rightarrow \chi_{Q_i} \nabla u_0\) and \(\chi_{Q_{\varepsilon_ki}}A^\varepsilon \nabla u^{\varepsilon_k}\rightarrow \chi_{Q_i} A^0\nabla u_0\) weakly in \((L^2(Q))^N\), \(i=1,2\) (here \(Q_1=\{x\in Q: x_N>0\}\), \(Q_2=\{x\in Q: x_N<0\}\) and \(A^0\) is the homogenized tensor). Properties of \(u_0\) on varying of the parameters \(\gamma\) and \(\kappa\) are also described. Finally, a corrector matrix \(C^\varepsilon\) for the \(L^1\)-convergence to 0 of the sequence \(u^{\varepsilon_k}-C^{\varepsilon_k}\nabla u_0\) is explicitly computed.NEWLINENEWLINEMain ingredients for the proofs of the existence and regularity results are some a priori estimates and a truncation technique yielding a sequence of regularized problems which approximates \((P_\varepsilon)\). For the proof of the homogenization result, the main tool is a theorem stating that for \(\varepsilon_k\) small enough, the gradient of the solution \(u^{\varepsilon_k}\) is equivalent to that of a suitable linear problem.