Calculus of Variations and Geometric Measure Theory

L. Koch - M. Ruf - M. Schäffner

On the Lavrentiev gap for convex, vectorial integral functionals

created by ruf on 26 Jun 2023



Inserted: 26 jun 2023

Year: 2023

ArXiv: 2305.19934 PDF


We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W0{1,1}(\Omega)m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int\Omega W(x,\mathrm{D} u)\,\mathrm{d}x, $$ 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.