OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

In this way the functional J is l.s.c. for theweak* convergence of measures. If we furtherassume that f(s) > 0 for s > 0 andg(1) > 0, then we have J ≥ c > 0. The existencetheorem then applies and, given twoprobabilities µ 0 and µ 1 , we obtain at leasta minimizing path for the problem}min{J (γ) : γ(0) = µ 0 , γ(1) = µ 1 .provided J is not identically +∞.We now study two special cases (Ω ⊂ R Ncompact and m = dx):Concentration f ≡ +∞, g(z) = |z| rr ∈]0, 1[. We have then∫J(µ) = |µ(x)| r d#Ωµ atomicwithDiffusion f(z) = |z| q with q > 1, g ≡ +∞.We have then∫J(µ) = |u(x)| q dx µ = u · dx, u ∈ L q .Ω56