A \(\Gamma\)-convergence result for optimal design problems
The author considers optimal design models of a two-phase elastic mixture described by functionals \begin{align*} &I_p(\chi,u)=\left(\int_\Omega\left(\chi W_1(\nabla u)^p+(1-\chi)W_2(\nabla u)^p\right)\,dx\right)^{\frac 1p},\\ & \text{if}\ (\chi,u)\in L^\infty(\Omega;\{0,1\})\times W^{1,p}(\Omega;\mathbb{R}^m),\ \text{or} +\infty\ \text {otherwise}, \end{align*} where \(p\equiv \{p_n\},\ 1<p_n\to +\infty\) and \(W_i^p,\ i=1,2\), models the energy density of the \(i\)th phase. The functionals represent the elastic energy of the solid occupying the domain \(\Omega\) and undergoing the deformation \(u\), while \(\chi\) represents the characteristic functions of the first phase of stiffness or electric resistivity. The main result of the paper is the \(\Gamma\)-convergence of functionals \((I_p)_{p>p_0}\) as \(p\) goes to \(+\infty\) with respect to the \(L^\infty(\Omega;\{0,1\})\) weak-star and \(W^{1,p_0}(\Omega;\mathbb{R}^m)\) weak topology.
scientific article; zbMATH DE number 7600083
scientific article