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 byJ(u)=∫Ω×ΩW(∇u(x),∇u(y))dxdy, is weakly lower semicontinuous in W1,p(Ω)W1,p(Ω) if and only if W is separately convex. We assume that Ω   is an open set in RnRn 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.