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This pre-print has now been superseded by arXiv:2305.19934 and will not bepublished.  We prove that for convex vectorial functionals with (p,q)-growth theLavrentiev phenomenon does not occur up to the boundary when (p,q) are suitablyrestricted. Under minimal assumptions on the regularity of the domain and theboundary data, we obtain results for autonomous and non-autonomous functionals,under natural, controlled and controlled duality growth bounds.

On global absence of Lavrentiev gap for functionals with (p,q)-growth