Published Paper

Inserted: 22 jun 2024
Last Updated: 22 jun 2024

Journal: Advances in Nonlinear Analysis
Volume: 13
Number: 1
Year: 2024
Doi: https://doi.org/10.1515/anona-2024-0002

Abstract:

We consider the functional $ \mathcal{F}(u) := \int_\Omega f(x,Du(x))\ dx, $ where $f(x,z)$ satisfies a $(p,q)$-growth condition with respect to $z$ and can be approximated by means of a suitable sequence of functions. We consider $B_R\Subset\Omega$ and the spaces $ \label{spaces} X=W^{1,p}(B_R, \mathbb{R}^N)\quad\text{ and }\quad Y=W^{1,p}(B_R, \mathbb{R}^N)\cap W^{1,q}_{\text{loc}}(B_R,\mathbb{R}^N). $ We prove that the lower semicontinuous envelope of $\mathcal{F}
_Y$ coincides with $\mathcal{F}$ or, in other words, that the Lavrentiev term is equal to zero for any admissible function $u\in W^{1,p}(B_R, \mathbb{R}^N)$.

Keywords: regularity, minimizer, Lavrentiev’s phenomenon