Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.