On the Lavrentiev gap for convex, vectorial integral functionals

TL;DR

This paper proves the absence of the Lavrentiev gap for certain convex vectorial integral functionals under specific growth and regularity conditions, extending understanding in the calculus of variations.

Contribution

It establishes conditions under which the Lavrentiev gap does not occur for non-autonomous convex integral functionals with particular growth constraints.

Findings

No Lavrentiev gap for p ≤ d-1 with q-growth conditions.

Unbounded integrands allowed for p > d-1.

Growth conditions depend on the regularity of boundary data.

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.