OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

Variational integrals w.r.t. a measureConsider the functional∫F (u) = f(x, Du) dµdefined for smooth functions u (and +∞elsewhere), where µ is a general nonnegativemeasure.On the integrand f we assumethe usual convexity and p-growth conditions.We define the tangent fieldsX p′µ = { φ ∈ L p′µ (R n ; R n ) :the tangent spacesdiv(φµ) ∈ L p′µ (R n ) } ,T p µ(x) = µ-ess ⋃ {φ(x) : φ ∈ Xp ′µthe tangential gradient (for smooth functions)D µ u(x) = P µ (x, Du(x)).},21