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.