Stochastic homogenization of fractional obstacle problems

TL;DR

This paper establishes a stochastic homogenization framework for nonlinear, nonlocal obstacle problems involving fractional p-Laplacian operators with randomly distributed obstacles, extending previous periodic and well-separated models.

Contribution

It introduces a novel homogenization approach for complex obstacle distributions, including clustering effects and random shapes, using Palm measures and identifying a critical scaling regime.

Findings

Identified a critical scaling where obstacle capacity density is additive almost surely.

Derived an effective homogenized problem analogous to periodic or well-separated cases.

Extended analysis to obstacles with random shapes and nonlocal interaction kernels.

Abstract

We prove a stochastic homogenization result for a class of \emph{nonlinear} and \emph{nonlocal} variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the \emph{fractional} $p$-Laplacian, $p>1$, where a pinning condition $u=0$ is imposed on the solution in a \emph{random} collection of small balls whose centers and radii are generated by a \emph{stationary marked point process}. Such a general obstacle distribution allows for \emph{clustering effects} to appear with positive probability. Under suitable moment conditions on the obstacle radii, we identify a critical scaling regime in which the fractional $p$-capacity density of the obstacles is asymptotically additive \emph{almost surely}. In turn, this key property allows us to derive an effective homogenized problem which is formally analogous to the one obtained in the periodic setting or under the assumption of well-separation for the obstacles. The analysis also extends to the case of \emph{randomly shaped obstacles} and to a broad class of \emph{nonlocal interaction kernels}. At the methodological level, the paper develops a streamlined proof strategy with several new ingredients, among them the use of Palm measures.