Density of Smooth Functions in Sobolev-Orlicz Spaces

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Anisotropic parabolic equations with variable nonlinearity (2009, background): “0 () can be defined then as the closure of C 1 0 () with respect to the norm (2.2) ‐ see [41, 44 ].”
Some Approximation Properties in Musielak-Orlicz-Sobolev Spaces (2012, result): “…m 0 L ϕ (Ω) for the modular convergence, under the Our result generalizes that of the authors in [2] in the case of Musielak-Orlicz-Sobolev spaces, that of Gossez in [3] in the case of classical Orlicz spaces and those of Zhikov [4] [5] and Samko [6] in the case of variable exponent Sobolev spaces.”
Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, background): “9), involving iterated logarithms, has been found by Zhikov for the proof of first density of smooth functions and then higher integrability results [218,219].”
Minimizers of the variable exponent, non-uniformly convex Dirichlet energy (2008, unknown): “Density of smooth functions is investigated in [24,40,50,53], and a basic result is that density holds at least if the exponent is log-Hölder continuous.”
Solvability of the three-dimensional thermistor problem (2008, background): “It is also known (see [10]) that the set C∞ 0 (Ω) is dense in W if the exponent p(x) has a logarithmic modulus of continuity, in particular if p(x) is Hölder continuous in Ω, e.”