Quasi-Convex Integrands and Lower Semicontinuity in L¹

In this paper it is shown that, under mild continuity and growth hypotheses, if $f(x,u,.)$ is quasi-convex and if uₙ, u ∈ W1,1 are such that uₙ → u in L¹, then \[ ∫_Ω {f(x,u(x),∇ u(x))dx lim inf } ∫_Ω {f( {x,u_n (x),∇ u_n (x)} )dx.} \] The proof relies on a blowup argument in connection with a truncation result that allows one to consider uniformly convergent sequences.