We characterize necessary and sufficient conditions ensuring approximation by smooth maps in the Sobolev space W1,φ[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}W1,φ{document} between suitable compact manifolds, where the prototypical example of Young function φ[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}φ{document} is the double-phase integrand.
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Antonini et al. (2026) studied this question.
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