Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k ⩾ 3 be an integer. For 0< s <1, let D s ⊂ R 2 be the set that is constructed iteratively as follows. Take a regular open k ‐gon with sides of unit length, attach regular open k ‐gons with sides of length s to the middles of the edges, and so on. At each stage of the iteration the k ‐gons that are added are a factor s smaller than the previous generation and are attached to the outer edges of the family grown so far. The set D s is defined to be the interior of the closure of the union of all the k ‐gons. It is easy to see that there must exist some s k > 0 such that no k ‐gons overlap if and only if 0 < s ⩽ s k . We derive an explicit formula for s k . The set D s is open, bounded, connected and has a fractal polygonal boundary. Let E D s( t ) denote the heat content of D s at time t when D s initially has temperature 0 and ∂ D s is kept at temperature 1. We derive the complete short‐time expansion of E D s( t ) up to terms that are exponentially small in 1/ t . It turns out that there are three regimes, corresponding to 0< s <1/( k −1), s =1/( k −1), and 1/( k −1)< s ⩽ s k . For s ≠ 1/( k −1) the expansion has the form E D s ( t ) = p s ( log ⁡ t ) t 1 − d s / 2 + A s t 1 / 2 + B t + O ( e − r s / t ) ,where p s is a log (1/ s 2 )‐periodic function, d s =log ( k −1)/log (1/ s ) is a similarity dimension, A s and B are constants related to the edges and vertices, respectively, of D s , and r s is an error exponent. For s =1/( k −1), the t 1/2 ‐term carries an additional log t . 1991 Mathematics Subject Classification : 11D25, 11G05, 14G05.