Geodesic
On density of smooth functions in Sobolev--Orlich spaces
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 67-81

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We study the question of density of smooth functions in Sobolev spaces of variable orders also called Sobolev–Orlich spaces. In particular, we generalize the earlier obtained logarithmic condition and give a number of examples. 
@article{ZNSL_2004_310_a3,
     author = {V. V. Zhikov},
     title = {On density of smooth functions in {Sobolev--Orlich} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--81},
     publisher = {mathdoc},
     volume = {310},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_310_a3/}
}
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V. V. Zhikov. On density of smooth functions in Sobolev--Orlich spaces. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 67-81. http://geodesic.mathdoc.fr/item/ZNSL_2004_310_a3/