OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

In the mass optimization problem there ismore freedom than in the shape optimizationproblem because we may use measuresdifferent from µ = 1 Ω dx, as it occurs inshape optimization problems.On the other hand, we are not restrictedonly to forces in H −1 or to n or n−1 dimensionalDirichlet regions; for instance we mayconsider forces and Dirichlet regions concentratedin a single point, or in a finite numberof points.Of course, for a given f, we may haveE(µ) = −∞ for some measures µ; for instancethis occurs when f is a Dirac massand µ is the Lebesgue measure on a domainΩ, but these measures with infinite complianceare ruled out by the optimization criterionwhich consists in maximizing E(µ).15