Optimal design and constrained quasiconvexity (Q2706215)

The author considers an optimal design problem of the form NEWLINE\[NEWLINE\text{Minimize }I(g) = \int_\Omega W(x,g(x),w(x), \nabla w(x)) dx,NEWLINE\]NEWLINE with \(g(x) \in \{a,b\}\), \({1\over |\Omega|} \int_\Omega g = \lambda a + (1 - \lambda)b\), \(\lambda \in(0,1)\) is fixed, \(w\in H^1_0(\Omega)\) is the solution of -div\((g \nabla w)=f\) for a given \(f \in H^{-1}(\Omega)\), \(\Omega\) is a domain in \(R^2\), and the integrand \(W\) is a Carathéodory function which does not satisfy any convexity property. In general the above problem has no classical solutions. The author is interested in a relaxed formulation. For this, he shows the equivalence of this problem with a vector variational problem of the form NEWLINE\[NEWLINE\text{Minimize }\int_\Omega \varphi(x,u(x), \nabla u(x)) dx,NEWLINE\]NEWLINE where \(u = (u^1,u^2)\in H^1(\Omega;R^2)\), \(u^1 \in H^1_0(\Omega)\) and \(\int_\Omega \psi(x,u(x), \nabla u(x)) dx = \lambda\), for appropriate densities \(\varphi\) and \(\psi\). The corresponding relaxed problem is defined thanks to \(H^1\)-Young measures.