Relaxation through homogenization for optimal design problems with gradient constraints
The paper considers optimal control problems for the elliptic equation \[ \text{div}\Biggl(\sum^N_{s=1} \theta_s(x) A_s\nabla u(x)\Biggr)= f\quad\text{in }\Omega,\quad u\in H^1_0(\Omega),\quad\Omega\subset \mathbb{R}^n,\tag{1} \] where \(A_s\), \(s= 1,\dots, N\), are given positive definite symmetric \(n\times n\)-matrices, the control \(\theta\) belongs to the set \(S\), \[ \begin{multlined}\theta\in S= \Biggl\{\theta\in L_\infty(\Omega;\mathbb{R}^N)|\theta= (\theta_1,\dots, \theta_N),\quad \theta_1(x)= 0\quad\text{or }1,\\ \int_\Omega \theta_s(x) dx\leq \gamma_s,\;s= 1,\dots, N,\;\theta_1(x)+\cdots+ \theta_N(x)= 1\Biggr\},\end{multlined} \] and on the state \(u\) are imposed constraints of the type \[ \int_\Omega p(x)|\nabla u(x)|^2 dx\leq d.\tag{2} \] The author introduces a notion of \(G\)-convergence of sequences of mappings \(A_k(\cdot,x)\), parameterized by \(x\in\Omega\), \[ A_k(\cdot, x)(B_1,\dots, B_N)= \sum^N_{s=1} \theta^k_s(x) B_s,\quad \theta^k\in S,\quad k= 1,2,\dots, \] and by using local correctors shows that the limit mapping \(A_0(\cdot, x)\) is differentiable (in an appropriate sense) at \((A_1,\dots, A_N)\). Obtained expressions for these derivatives allow him to show that \[ \int_\Omega p(x)|\nabla u_k(x)|^2 dx\to \int_\Omega p(x)\langle(\sigma(x)+ I) \nabla u_0(x), \nabla u_0(x)\rangle dx\quad\text{as }k\to \infty, \] where \(\{u_k\}\) is the sequence of solutions of (1) with \(\theta= \theta^k\), \(k= 1,2,\dots, u_k\to u_0\) weakly in \(H^1_0(\Omega)\) and the matrix \(\sigma\) is explicitly defined by derivatives of the limit mapping \(A_0\) at \((A_1,\dots, A_N)\). After that the author formulates a relaxed problem, shows the existence of a solution \(u_0\) for this problem and establishes the existence of a sequence \(\{\theta^k\}\subset S\) for which the corresponding solutions \(u_k\) of (1) with \(\theta= \theta^k\), \(k= 1,2,\dots\), converge weakly to \(u_0\) and satisfy (2) with \(d+ 1/k\) instead of \(d\).
scientific article; zbMATH DE number 1812477