Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

In this work we are going to prove the functional J defined by J ( u ) = ∫ Ω × Ω W ( ∇ u ( x ) , ∇ u ( y ) ) d x d y , is weakly lower semicontinuous in W 1 , p ( Ω ) if and only if W is separately convex. We assume that Ω is an open set in R n and W is a real-valued continuous function fulfilling standard growth and coerciveness conditions. The key to state this equivalence is a variational result established in terms of Young measures.