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Characterization of optimal shapes and masses through Monge-Kantorovich equation. (English) Zbl 0982.49025
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 $J(\mu, u) = \int j(Du) d\mu$ 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.
The authors discuss existence of relaxed solutions of this problem and the related necessary and sufficient conditions for optimality.}

##### MSC:
 49Q10 Optimization of shapes other than minimal surfaces 49J45 Methods involving semicontinuity and convergence; relaxation 49J20 Existence theories for optimal control problems involving partial differential equations 90B06 Transportation, logistics and supply chain management 28A50 Integration and disintegration of measures 49K20 Optimality conditions for problems involving partial differential equations
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