Submitted Paper

Inserted: 10 apr 2026
Last Updated: 10 apr 2026

Year: 2026

Abstract:

We prove a stochastic homogenization result for a class of nonlinear and nonlocal variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the fractional $p$-Laplacian, $p>1$, where a pinning condition $u=0$ is imposed on the solution in a random collection of small balls whose centers and radii are generated by a stationary marked point process. Such a general obstacle distribution allows for 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 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 randomly shaped obstacles and to a broad class of 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.

Keywords: $\Gamma$-convergence, Nonlocal functionals, obstacle problems, stochastic homogenization, fractional capacity, stationary processes, marked point processes , Palm measures

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