OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

The operator D µ is closable, i.e.{uh → 0 weakly in L p µD µ u h → v weakly in L p ⇒ v = 0 µa.e.µand we still denote by D µ its closure. TheSobolev space W 1,pµ is then defined as thedomain of the extension above, with norm‖u‖ 1,p,µ = ‖u‖ Lpµ+ ‖D µ u‖ Lpµ.The relaxation of the integral functionalabove is given by∫F (u) = f µ (x, D µ u) dµ u ∈ Wµ1,pwheref µ (x, z) = inf { f(x, z+ξ) : ξ ∈ ( T p µ(x) ) ⊥}.In particular,∫|Du| 2 dµrelaxes to∫|D µ u| 2 dµ .22