Non-occurrence of gap for one-dimensional autonomous functionals

Let $F(u) := \int_0^1 f(u,u') dt$ be a weakly lower semicontinuous autonomous functional defined for all functions $u:[0,1]\to R^n$ in the Sobolev space $W^{1,p}$. We show that under suitable hypotheses $F$ agrees with the relaxation of the same functional restricted to regular functions, i.e., that for every function $u$ there exist regular functions $u_h$ such that $u_h\to u$ in the $W^{1,p}$ norm and $F(u_h) \to F(u)$