Fractal relaxed Dirichlet problems
Let \(K\) be a self-similar fractal with equal ratio \(\rho\) for its \(N\) similitudes \(S_ 1,\dots,S_ N\) and Hausdorff dimension \(\alpha>n-2\) \((N \rho^ \alpha=1)\). The authors construct first for each \(p \in \mathbb{N}\) a closed subset \({\mathcal E}_ p\) mainly depending on the ingredients \(\rho\), \(S_ 1,\dots,S_ N\) leading to \(K\). After that they consider for \(f \in L^ 2(\Omega)\), where \(\Omega\) is a bounded open subset of \(\mathbb{R}^ n\), the Dirichlet problem \(-\Delta u_ p=f\) in \(\Omega \backslash {\mathcal E}_ p\) and \(u_ p \in H^ 1_ 0 (\Omega \backslash {\mathcal E}_ p)\). The main result is that the weak solution \(u_ p\) converges in \(L^ 2(\Omega)\) to the weak solution of the problem \(-\Delta u+c_ 1u {\mathcal H}^ \alpha |_{K \cap \Omega}=f\) in \(\Omega\), \(c_ 1\) a well-defined constant, \({\mathcal H}^ \alpha\) the \(\alpha\)-dimensional Hausdorff measure, and \(u \in H^ 1_ 0 (\Omega)\). This reads more comprehensible as \[ \int_ \Omega Du Dv dx+c_ 1 \int_{K \cap \Omega} uv d {\mathcal H}^ \alpha=\int_ \Omega fv dx \] for every \(v \in H^ 1_ 0 (\Omega)\).
scientific article; zbMATH DE number 510012