On the Lavrentiev gap for convex, vectorial integral functionals

We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form F : g + W 0 1 , 1 ( Ω ) m → [ 0 , + ∞ ] , F ( u ) = ∫ Ω J ( x , D u ) d x , where the boundary datum g : Ω ⊂ R d → R m is sufficiently regular, ξ ↦ J ( x , ξ ) is convex and lower semicontinuous, satisfies p -growth from below and suitable growth conditions from above. More precisely, if p ≤ d − 1 , we assume q -growth from above with q ≤ ( d − 1 ) p d − 1 − p , while for p > d − 1 or p = 1 if d = 2 , we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the x -dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions.