The authors consider the energy functional \[ {\mathcal E}(\mu) = \inf_u \bigg( \int j(Du) d\mu - \langle f, u \rangle \bigg) \] where \(\mu\) is a nonnegative measure with support in \(\overline \Omega\) \((\Omega\) an \(n\)-dimensional domain). The function \(j\) (the stored energy density function) is defined on \(n \times n\) matrices, \(f\) is a bounded vector valued measure and the infimum is taken over smooth displacements \(u\) of the \(n\)-dimensional space into itself that vanish on a closed subset \(\Sigma \subseteq \overline \Omega.\) The compliance is \({\mathcal C}(\mu) = - {\mathcal E}(\mu),\) and the mass optimization problem is to minimize \({\mathcal C}(\mu)\) with constraints \(\int d \mu = m,\) support \(\mu \subseteq \overline \Omega.\)
Under assumptions where the solutions of the mass optimization problem may not be unique, the authors construct a particular solution by means of an approximation involving the \(p\)-Laplace equation.