Non occurrence of the Lavrentiev gap for a class of nonautonomous functionals

We consider a multidimensional scalar problem of the calculus of variations with a nonnegative general Lagrangian depending on the space variable, on a Sobolev function and on its gradient. The main result of the paper is a sufficient condition discarding the Lavrentiev phenomenon between Sobolev and Lipschitz functions, with a prescribed boundary datum. The result unifies most of the known conditions in the literature and does not require that the Lagrangian obey a p-q growth condition or be an N-function.