The $p$-Laplacian in domains with small random holes

Balzano, M. ; Durante, T.

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 435-458

Résumé

We investigate sequences of nonlinear Dirichlet problems of the form \begin{equation*} \tag{$P_h$} \begin{cases} -\text{div}\,(|Du_{h}|^{p-2} Du_{h})=g, \ \text{in } D \setminus E_{h} \\ u_{h}\in H^{1,p}_{0}(D \setminus E_{h}). \end{cases} \end{equation*} where $2\leq p \leq n$ and $E_{h}$ are random subsets of a bounded open set $D$ of $\mathbb{R}^{n}$$(n\geq 2)$. By means of a variational approach, we study the asymptotic behaviour of solutions of $(P_h)$, characterizing the limit problem for suitable sequences of random sets.