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

TL;DR

This paper investigates the convergence of nonlinear integral functionals in randomly perforated domains, establishing conditions under which the limit problem is well-defined and free of percolation, even with large perforations.

Contribution

It introduces a minimal assumption framework for the size of perforations in random domains, extending previous deterministic results to stochastic settings with finite moments.

Findings

Limit problem is well-defined under finite $(n-q)$-moment condition.

Critical rescaling prevents percolation in the limit.

Averaged nonlinear capacitary term derived for random perforations.

Abstract

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 \emph{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 \cite{Ansini-Braides}. In analogy to the random setting introduced by Giunti, H\"ofer, and Vel\'azquez \cite{Giunti-Hofer-Velasquez} 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.