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 $$I(u)=\int_{\Omega}F(x, u(x), \nabla u(x))\,dx$$ where $\Omega$ is a bounded open subset of $\mathbb{R}^n$ and $F\colon \Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$ is a given Carathéodory function; the admissible functions $u$ coincide with a given Lipschitz function on $\partial\Omega$. We formulate some conditions under which a given function in $\phi+W^{1,p}_0(\Omega)$ with $I(u)<+\infty$ can be approximated by a sequence of functions $u_k\in\phi+W^{1,p}_0(\Omega)\cap L^{\infty}$ converging to $u$ in the norm of $W^{1,p}$, and such that $I(u_k)\rightarrow I(u)$. The problem is strictly related with the non occurrence of the Lavrentiev gap.