Non-Occurrence of a Gap Between Bounded and Sobolev Functions for a Class of Nonconvex Lagrangians

We consider the classical functional of the Calculus of Variations of the form I(u) = ZΩ F(x, u(x), ∇u(x)) dx where Ω is a bounded open subset of Rn and F : Ω×R×Rn → R is a given Carathéodory function; the admissible functions u coincide with a given Lipschitz function on ∂Ω. We formulate some conditions under which a given function in ϕ + W01,p(Ω) with I(u) < +∞ can be approximated by a sequence of functions uk ∈ ϕ+W01,p(Ω)∩L∞ converging to u in the norm of W1,p, and such that I(uk) → I(u). The problem is strictly related with the non occurrence of the Lavrentiev gap.