Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces

We prove the density of smooth functions in the modular topology in Musielak–Orlicz–Sobolev spaces essentially extending the results of Gossez [17] obtained in the Orlicz–Sobolev setting. We impose new systematic regularity assumption on M which allows to study the problem of density unifying and improving the known results in Orlicz–Sobolev spaces, as well as variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of functions by smooth functions in the double-phase space governed by the modular function with excluding the Lavrentiev phenomenon within the sharp range . See [11, Theorem 4.1] for the sharpness of the result.