On the Lavrentiev gap for convex, vectorial integral functionals

We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$F:g+W_0^{1,1}(\mathrm{\Omega }{)}^m\to \mathbb{R}\cup \{+\mathrm{\infty }\},F(u)={\int }_{\mathrm{\Omega }}W(x,\mathrm{D}u)\mathrm{d}x,$$ where the boundary datum $g:\mathrm{\Omega }\subset {\mathbb{R}}^d\to {\mathbb{R}}^m$ is sufficiently regular, $\xi \mapsto W(x,\xi )$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\le d-1$, we assume $q$-growth from above with $q\le \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ 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.