On the Lavrentiev gap for convex, vectorial integral functionals

Abstract

We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form$F:g+W_0^{1,1}{\left(\Omega \right)}^m\to [0,+\infty ],F\left(u\right)=\underset{\Omega }{\int }J(x,Du)dx,$ where the boundary datum $g:\Omega \subset R^d\to R^m$ is sufficiently regular, $\xi \mapsto J(x,\xi )$ 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 $p>d-1$ 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.