Renewal equation for the heat equation of an arithmetic von Koch snowflake (Q2759732)

This paper investigates the asymptotic behaviour of the heat content as time \(t\to 0+\) for an arithmetic von Koch snowflake generated by a regular \(k\)-gon, \(K_{k,s}\) constructed as follows: Fix an integer \(k\geq 3\) and let \(V_j={1\over 2}(\text{cosec}{\pi \over k})e{2\pi ji\over k}\), \(j=1,\dots,k\), be the vertices of a regular \(k\)-gon with volume \({k\over 4}\text{cot}{\pi\over k}\) and boundary of length \(k\). Fix \(s\in (0,1)\). \(\partial K_{k,s}\) is constructed by repeatedly replacing the middle portion \(s\) of each segment, beginning with \(V_1V_2\), \(V_2V_3,\dots, V_{k-1}V_k\), and \(V_kV_1\) by the \(k-1\) other sides of a regular \(k\)-gon. The results include those known for \(k=3\), \(s={1\over 3}\) and \(k=4\), \(s\in A_4\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00044].