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...

We give sufficient conditions for the continuity in norm of the translation operator in the Musielak-Orlicz LM spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in LM , in both modular and norm topologies. These density results are then applied to obtain basic topological properties. 1 2 A. YOUSSFI AND Y. AHMIDA Define M : Ω × [0, ∞) → [0, ∞] by M (x, s) = sup t 0 {st − M (x, t)} for all s 0 and all x ∈ Ω. It can be checked that M ∈ φ. The Φ-function M is called the complementary function to M in the sense of Young. Given M ∈ φ, the Musielak-Orlicz space L M (Ω) consists of the set of all measurable functions u : Ω → R such that Ω M (x, |u(x)|/λ)dx < +∞ for some λ > 0. Equipped with the so-called Luxemburg norm [11, Theorem 7.7]). It is a particular case of the so-called modular function spaces, investigated by H. Nakano (see for instance [12]). We define E M (Ω) as the subset of L M (Ω) of all measurable functions u : Ω → R such thatDensity result of smooth functions in Musielak-Orlicz-Sobolev spaces with respect to the modular topology was claimed for the first time in [2] in Ω = R N and then for a bounded star-shape Lipschitz domain Ω in [3]. The authors assumed that the Φ-function M satisfies, among others, the log-Hölder continuity condition, that is to say there exists a constant A > 0 such that for all s 1,

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We give a sufficient structural condition on the Musielak functions for the Meyers–Serrin theorem to hold in Musielak spaces. We unify and extend some density results of smooth functions in Orlicz–Sobolev spaces as well as in variable exponent Sobolev spaces.

In this paper we investigate Poincaré-type integral inequalities in the functional Musielak structure. We extend the ones already well known in Sobolev, Orlicz and variable exponent Sobolev spaces. We introduce conditions on the Musielak functions under which they hold. The identification with null trace functions space is given. Ω |α| 0 is a constant. We also get the same inequality in the subspace W m 0 E M (Ω) under minimal assumptions.