Calculus of Variations and Geometric Measure Theory

C. D'Onofrio - C. I. Zeppieri

$\Gamma$-convergence and stochastic homogenisation of degenerate integral functionals in weighted Sobolev spaces

created by zeppieri1 on 15 Sep 2021
modified on 03 Jan 2022


Accepted Paper

Inserted: 15 sep 2021
Last Updated: 3 jan 2022

Journal: Proc. Roy. Soc. Edinburgh Section A
Year: 2022


We study the $\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space.

The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenisation result for the corresponding functionals.

Keywords: $\Gamma$-convergence, weighted Sobolev spaces, Muckenhoupt weights, stochastic homogenisation, degenerate growth conditions