Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals
This interesting paper is devoted to the relaxation of the problem \[ I= \int_\Omega|\nabla u(x)-\overline g(x)|^2 dx\to \min, \] \[ \text{div}(a(x)\nabla u(x))= f(x)\quad\text{in }\Omega,\quad u\in H^1_0(\Omega),\tag{1} \] \[ a\in U_\gamma= \Biggl\{a(x)= \alpha\chi(x)+ \beta(1-\chi(x)), \chi(x)= 0\text{ or }1,\;\int_\Omega\chi(x) dx= \gamma|\Omega|\Biggr\}, \] where \(\Omega\subset \mathbb{R}^n\), \(n= 2,3\), is a bounded Lipschitz domain and \(\alpha< \beta\), \(0< \gamma< 1\) are given positive constants. The crux of the problem is that \(I\) is not weakly continuous, hence, a straightforward passage to the \(G\) (or \(H\))-closure of the set \(U_\gamma\) does not give a relaxation of (1). In the first part of the paper the author shows that the relaxed problem is \[ I^*= \int_\Omega \Biggl[|\overline g(x)|^2- 2\langle\overline g(x),\nabla u(x)\rangle- {1\over\beta} f(x) u(x)+ {|(\beta I- A(x))\nabla u(x)|^2\over \beta(\beta- \alpha)\theta(x)}\Biggr] dx\to \min, \] \[ \text{div}(A(x)\nabla u(x))= f(x)\quad\text{in }\Omega,\quad u\in H^1_0(\Omega),\tag{2} \] \[ \theta\in S_\gamma= \Biggl\{\theta\text{ measurable},\;0\leq\theta(x)\leq 1,\;\int_\Omega \theta(x) dx= \gamma|\Omega|\Biggr\}, \] \[ A(x)\in G_{\theta(x)}\quad\text{a.e. }x\in\Omega, \] where \(I\) is the unit matrix and \(G_\theta\) is the so-called \(G_\theta\)-closure which represents all effective \(n\times n\)-matrices of periodic composites with prescribed volume fraction \(\theta\) of the material \(\beta\). In the second part of the paper it is shown that if in (2) \(\theta\in \bigcup S_\gamma\), \(0\leq\gamma\leq 1\) (i.e., there are no constraints on the volume fractions), then the minimizers \(A_0\) in (2) are rank-one laminates.
scientific article; zbMATH DE number 1763532
scientific article