Gossez's approximation theorems in Musielak-Orlicz-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 W01,p(Ω) functions by smooth functions in the double-phase space governed by the modular function H(x,s)=sp+a(x)sq with a∈C0,α(Ω) excluding the Lavrentiev phenomenon within the sharp range q/p≤1+α/N. See [11, Theorem 4.1] for the sharpness of the result.