*preprint*

**Inserted:** 26 jun 2023

**Year:** 2023

**Abstract:**

We prove the absence of a Lavrentiev gap for vectorial integral functionals
of the form
$$
F: g+W_{0}^{{1,1}}(\Omega)^{m\to\mathbb{R}\cup\{+\infty\},\qquad} F(u)=\int_{\Omega}
W(x,\mathrm{D} u)\,\mathrm{d}x,
$$
where the boundary datum $g:\Omega\subset \mathbb{R}^d\to\mathbb{R}^m$ 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 $p\leq d-1$, we assume $q$-growth from above with $q\leq
\frac{(d-1)p}{d-1-p}$, 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.