Asymptotics of the heat exchange

Let K be a compact subset in Euclidean space Rm, and let EK(t) denote the total amount of heat in Rm⧹K at time t, if K is kept at fixed temperature 1 for all t⩾0, and if Rm⧹K has initial temperature 0. For two disjoint compact subsets K1 and K2 we define the heat exchange HK1,K2(t)=EK1(t)+EK2(t)−EK1∪K2(t). We obtain the leading asymptotic behaviour of HK1,K2(t) as t→0 under mild regularity conditions on K1 and K2.