Summary
LetD be an open, bounded set in euclidean space ℝm (m=2, 3, ...) with boundary ∂D. SupposeD has temperature 0 at timet=0, while ∂D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE D (t), the amount of heat inD at timet, ast→0+. For the triadic von Koch snowflakeK our results imply that
for some constantc>1.
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van den Berg, M. Heat content and Brownian motion for some regions with a fractal boundary. Probab. Th. Rel. Fields 100, 439–456 (1994). https://doi.org/10.1007/BF01268989
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DOI: https://doi.org/10.1007/BF01268989