Fractal relaxed Dirichlet problems
type: Item
instance of: scholarly article
title: Fractal relaxed Dirichlet problems
zbMATH Open document ID: 0794.28004
DOI: 10.1007/BF02567843
author: Andrea Braides
author: Lino Notarantonio
published in: Manuscripta Mathematica
publication date: 1994-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/155886
review text: 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 \(v \in H^ 1_ 0 (\Omega)\).
reviewed by: Hermann Haase
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 510012
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: self-similar fractal
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Hausdorff measure
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 510012