Smooth functions are not dense in norm topology in Orlicz-Sobolev spaces generated by a convex function of arbitrary growth, but in the weaker one called modular topology. The same concerns their anisotropic versions. On the other hand, in the inhomogeneous setting of variable exponent spaces, one can prove density of Lipschitz or smooth functions only if the exponent is regular enough. Otherwise, the Lavrentiev phenomenon occurs and there exist functions that cannot be reasonably approximated.
The framework of anisotropic Musielak-Orlicz spaces in its full generality inherits both of these difficulties. I will discuss all known optimal conditions for density in sub-cases and how we cover them in the general case.
The talk is based on an extract from the series of recent joint papers
Ahmida, C., Gwiazda, Youssfi, JFA 2018; Gwiazda, Skrzypczak (C.), Zatorska-Goldstein, JDE 2018; Alberico, C., Cianchi, Zatorska-Goldstein, CalcVar 2019.