Calculus of Variations and Geometric Measure Theory

L. Scardia - K. Zemas - C. I. Zeppieri

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

created by zeppieri1 on 21 Jul 2023
modified by zemas on 13 Jan 2024


Submitted Paper

Inserted: 21 jul 2023
Last Updated: 13 jan 2024

Year: 2023


In this paper we study the convergence of integral functionals with $q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$. Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged analogue of the nonlinear capacitary term obtained by Ansini and Braides in the deterministic periodic case. In analogy to the random setting introduced by Giunti, Hoefer, and Velazquez to study the Poisson equation, we only require that the random radii have finite $(n-q)$-moment. This assumption on the one hand ensures that the expectation of the nonlinear $q$-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.