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.
Categories: math.AP