OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

More precisely, for every admissible domainΩ we consider the energy{ ∫E(Ω) = inf j(Du) dx − 〈f, u〉 :Ω}u smooth, u = 0 on Σand the elastic complianceC(Ω) = −E(Ω).In the linear case, when j is a quadraticform, an integration by parts gives that theelastic compliance reduces to the work ofexternal forcesC(Ω) = 1 2 〈f, u Ω〉being u Ω the displacement of minimal energyin Ω. In linear elasticity, if z ∗ = sym(z)and α, β are the Lamé constants,j(z) = β|z ∗ | 2 + α 2 |trz∗ | 2 .9