Density of compactly supported smooth functions C_C^∞(ℝ^d) in Musielak-Orlicz-Sobolev spaces W^1,Φ(Ω)

2023/05/17 by Anna Kamiśka, Kamiśka, Anna, Mariusz Żyluk +1
Mathematics · #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2305.10605

Abstract

We investigate here the density of the set of the restrictions from C_C^∞(ℝ^d) to C_C^∞(Ω) in the Musielak-Orlicz-Sobolev space W^1,Φ(Ω). It is a continuation of article \citeKamZyl3, where we have studied density of C_C^∞(ℝ^d) in W^{k, Φ}(ℝ^d) for k∈ℕ. The main theorem states that for an open subset Ω⊂ ℝ^d with its boundary of class C¹, and Musielak-Orlicz function Φ satisfying \rm condition (A1) which is a sort of log-Hölder continuity and the growth condition Δ₂, the set of restrictions of functions from C_C^∞(ℝ^d) to Ω is dense in W^1,Φ(Ω). We obtain a corresponding result in variable exponent Sobolev space W^1,p(⋅)(Ω) under the assumption that the exponent p(x) is essentially bounded on Ω and Φ(x,t) = t^p(x), t≥ 0, x∈Ω, satisfies the log-Hölder condition.

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