No Lavrentiev gap for some double phase integrals

We prove the absence of the Lavrentiev gap for non-autonomous functionalsF(u) := integral(Omega) f(x, Du(x)) dx,where the density f(x, z) is alpha-Holder continuous with respect to x is an element of Omega subset of R-n, it satisfies the (p, q)-growth conditionsvertical bar z vertical bar(p) <= f(x, z) <= L (1 + vertical bar z vertical bar(q)),where 1 < p < q < p(n+alpha/n), and it can be approximated from below by suitable densities f(k).

We prove the absence of the Lavrentiev gap for non-autonomous functionals F(u) := integral(Omega) f(x, Du(x)) dx,where the density f(x, z) is alpha-Holder continuous with respect to x; x is an element of Omega subset of R^n; f satisfies the (p, q)-growth conditions |z|^p <= f(x, z) <= L (1 + |z|^q),where 1 < p < q < p(n+alpha)/n, and it can be approximated from below by suitable densities f_k.