Characterization of optimal shapes and masses through Monge-Kantorovich equation (Q5940693)

The mass optimization problem consists of finding the best distribution of a given total mass in order to minimize elastic compliance under the action of a given force field \(f \in {\mathcal M}({\mathbf R}^n; {\mathbf R}^n) = \{{\mathbf R}^n\)-valued measures in \({\mathbf R}^n\) with finite total variation and compact support\(\}.\) A {displacement is a test function \(u(\cdot) \in {\mathcal D}({\mathbf R}^n; {\mathbf R}^n) =\{\)infinitely differentiable \({\mathbf R}^n\)-valued functions in \({\mathbf R}^n\) with compact support\(\}\) and \(j(Du)\) is the stored energy density associated with \(u,\) where \(j\) is a function satisfying various assumptions. For a given mass distribution \(\mu\) the stored elastic energy of \(u\) is NEWLINE\[NEWLINE J(\mu, u) = \int j(Du) d\mu NEWLINE\]NEWLINE and the problem is to minimize the total energy \(J(\mu, u) - \langle f, \mu \rangle\) over all measures having support in a given design region and all \(u\) possibly satisfying a boundary condition. NEWLINENEWLINENEWLINEThe authors discuss existence of relaxed solutions of this problem and the related necessary and sufficient conditions for optimality.}