Published Paper

Inserted: 23 jan 2024
Last Updated: 23 jan 2024

Journal: Advances in Calculus of Variations
Volume: 17
Number: 1
Pages: 165-194
Year: 2024
Doi: 10.1515/acv-2021-0109
Links: doi

Abstract:

We prove the absence of the Lavrentiev gap for non-autonomous functionals $ \mathcal{F}(u) := \int_{\Omega}f(x, Du(x)) \ dx, $ where the density $f(x,z)$ is $\alpha$-Hölder continuous with respect to $x \in \Omega \subset \mathbb{R}^n$, it satisfies the $(p,q)$-growth conditions $
z
^p \leq f(x,z) \leq L( 1 +
z
^q),$ where $1 <p < q < p \left (\frac{n+\alpha}{n} \right )$, and it can be approximated from below by suitable densities $f_k$.

Keywords: regularity, minimizer, variational, integral, lavrentiev, phenomenon