HEAT-CONTENT AND BROWNIAN-MOTION FOR SOME REGIONS WITH A FRACTAL BOUNDARY

Let D be an open, bounded set in euclidean space IR(m) (m = 2, 3,...) with boundary partial derivative D. Suppose D has temperature 0 at tim e t = 0, while partial derivative D is kept at temperature 1 for all t > 0. We use brownian motion to obtain estimates for the solution of c orresponding heat equation and to obtain results for the asymptotic be haviour of E(D)(t), the amount of heat in D at time t, as t --> 0(+). For the triadic von Koch snowflake K our results imply that c(-1) t(1- (log 2)/log 3) less than or equal to E(K)(t) less than or equal to ct( 1-(log 2)/log 3), 0 less than or equal to t less than or equal to c(-1 ) for some constant c > 1.