We prove the density of smooth functions in the modular topology in the Musielak-Orlicz-Sobolev spaces essentially extending the results of Gossez [16] 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 the Orlicz-Sobolev spaces, as well as the 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 W 1,p 0 (Ω) functions by smooth functions in the double-phase space governed by the modular function H(x, s) = s p + a(x)s q with a ∈ C 0,α (Ω) excluding the Lavrentiev phenomenon within the sharp range q/p ≤ 1 + α/N . See [10, Theorem 4.1] for the sharpness of the result. thermistor model [42]. Problems in various types of the Musielak-Orlicz spaces are widely considered from analytical point as well, inter alia the highly modern calculus of variations deals with minimization of the variational integrals [4,10,11,40] See also how the problem of minimisation is examined in the Musielak-Orlicz setting under ∆ 2 /∇ 2 -conditions [22]. The Lavrentiev phenomenonThe Musielak-Orlicz spaces do not inherit all the good properties of the classical Sobolev spaces. Besides reflexivity and separability, which are not -in general -the properties we deal with, the problems with density can also appear and is related to the so-called Lavrentiev phenomenon. We meet it when the infimum of the variational problem over the smooth functions is strictly greater than infimum taken over the set of all functions satisfying the same boundary conditions, cf. [30,40]. The notion of the Lavrentiev phenomenon became naturally generalised to describe the situation, where functions from certain spaces cannot be approximated by regular ones (e.g. smooth).It is known that in the case of the variable exponent spaces, the Lagrangian M (x, s) = |s| p(x) can exhibit the Lavrentiev phenomenon if p(·) is not regular enough (see e.g. [40, Example 3.2]). The canonical, but not optimal, assumption ensuring density of smooth function in norm topology in the variable exponent spaces is the log-Hölder continuity of the exponent p(·). The double-phase spaces (with M (x, s) = |s| p + a(x)|s| q ) can also support the Lavrentiev phenomenon [9,10,14], where the authors provide sharp result.The mentioned results show that the strong closure of the smooth functions can be not relevant type of useful approximation in the spaces with a not sufficiently regular modular function. We provide here sufficient conditions to avoid the Lavrentiev phenomenon. Let us point out that this type of result can be used in order to get e.g. regularity of minimisers cf. [10]. Approximation results in the Musielak-Orlicz spacesAn earlier density result of smooth functions in the Musielak-Orlicz-Sobolev spaces W m L M (R N ) with respect to the strong (norm) topology was proved first by Hudzik [26, Theorem 1] assuming the ∆ 2 -co...

The Musielak-Orlicz setting unifies variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity.In this survey we present an overview of developments of the theory of existence of PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev's phenomenon.

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N -function, which is not necessarily of power type and need not satisfy the ∆2 nor the ∇2condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutionsin the approximable sense -is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L 1 -data.