Perimeter on fractal sets
The authors remark that the Calculus of Variations and Geometric Measure Theory does not seem to have an easy extension to fractal problems. Their aim is to give a contribution to the problem defining lower semicontinuous functionals. They consider the following special case: Let \(V_ 0\) be an equilateral triangle of side length 1 with center in 0, \(V_{h+1}\) the union of \(V_ h\) and \(3\cdot 4^ h\) equilateral triangles of side length \(3^{-(h+1)}\) external to \(V_ h\), such that one of their edges lies on the central third of one of the \(3\cdot 4^ h\) edges of \(V_ h\). Then \(V=\overline{\cup V_ h}\) is the full snowflake whose boundary is composed of three copies of the Koch curve. Let denote \(\partial E\) the topological boundary of the set E and \(\partial^*E\) its essential boundary, that is the set \[ \partial^*E=\{x\in {\mathbb{R}}^ 2:\limsup_{\rho \to 0^+}\frac{| B_{\rho}(x)\setminus E|}{\rho^ n}>0,\quad \limsup_{\rho \to 0^+}\frac{| B_{\rho}\cap R|}{\rho^ n}>0\}. \] If A is any open set and E any set of finite perimeter the authors study the behaviour of the sequence of functionals \[ F_ h(E,A)={\mathcal H}_ 1(A\cap (\partial^*E\setminus \partial^*V_ h))+(3/4)^ h{\mathcal H}_{\alpha}(K){\mathcal H}_ 1(A\cap \partial^*E\cap \partial^*V_ h), \] where \({\mathcal H}_ 1,{\mathcal H}_{\alpha}\) are the 1-respectively \(\alpha\)-dimensional Hausdorff measure \((\alpha =\log 4/\log 3)\) and K stands for the Koch curve. Provided that E is measurable and \((\partial E\cap A)\setminus \partial^*V\) consists of a finite number of regular arcs then the so called \(\Gamma\)-limit of \((F_ n(E,A))\) in the sense of De Giorgi exists and is equal to \[ {\mathcal H}_ 1(A\cap (\partial^*E\setminus \partial^*V))+{\mathcal H}_{\alpha}(A\cap \partial^*E\cap \partial^*V) \] and thus becomes a lower semicontinuous functional.
scientific article; zbMATH DE number 4200524