Published Paper

Inserted: 23 jan 2024
Last Updated: 23 jan 2024

Journal: Rendiconti Lincei - Matematica e Applicazioni
Volume: 34
Number: 2
Pages: 451-463
Year: 2023
Doi: 10.4171/RLM/1014
Links: doi

Abstract:

The aim of this paper is to propose some results which we hope could contribute to understand better the Lavrentiev's phenomenon for energy integrals as $ F\left( u,\Omega \right) =\int_{\Omega }\left\{ f\left( x,Du\right) +\left \langle b\left( x\right) ,u\right \rangle \right\} \ dx, $ under some $p,q-$growth conditions as $c_{1}\left\vert z\right\vert ^{p}-c_{2}\leq f\left( x,z\right) \leq c_{3}\left\vert z\right\vert ^{q}+c_{4\,}$; in fact we expect that the Lavrentiev's phenomenon does not occur if the quotient $q/p$ is not too large in dependence of $n$, for instance as in the cases - either scalar or vectorial ones - that we consider in this manuscript.

Keywords: calculus of variations, Elliptic equations, Local minimizer, Elliptic systems, $p,q$-growth, Lavrentiev’s phenomenon