Summary: We study the lower semicontinuity of some functionals of the type \[ \int_\Omega f(x, u, Du) dx+ \int_{\overline\Omega} g(x, u)d\mu, \] where \(\Omega\subset \mathbb{R}^n\), \(f(x, s,\cdot)\) is a convex non- homogeneous function, and \(\mu\) is a Radon measure. In a particular case, an integral representation formula for the relaxed functional is proved.