Optimal design via variational principles: The three-dimensional case. (Q1414180)

The paper aims to start a systematic analysis of some three-dimensional optimization problems, generalizing some results of the same authors. The problems taken into account deal with the minimization of the integral \[ I(y,u)=\int_\Omega F(x,y(x),u(x),\nabla u(x))\,dx, \] subject to \(y\in L^\infty(\Omega;{\mathbb R}^n)\), \(y(x)\in K\) for a.e. \(x\in\Omega\), \(\text{ div}(G(x,y(x),u(x),\nabla u(x)))=0\) in \(\Omega\), boundary conditions on \(u\), and \(\int_\Omega H(x,y(x),u(x),\nabla u(x))\,dx\leq\alpha\). Here \(\Omega\subseteq{\mathbb R}^3\) is a regular domain, and \(K\) is a closed subset of \({\mathbb R}^n\). By the introduction of a suitable vector field \(V\), the problem is transformed into the minimization of \[ \widetilde I(u,V)=\int_\Omega \widetilde F(x,u(x),\nabla u(x),V(x))dx, \] subject to \(\text{div\,}V=0\) in \(\Omega\), and boundary conditions on \(u\). Hypotheses for the new variational problem are derived in order to provide existence results. Nonexistence and relaxation properties are also briefly analyzed. The underlying Young measures associated with the above divergence free constraint are also considered.