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Heat equation on the arithmetic von Koch snowflake

Abstract

We investigate the asymptotic behaviour of the heat content as the time t→ 0 for an s-adic von Koch snowflake generated by a square. We show that the heat content satisfies a functional equation which, after appropriate transformations, takes the form of an inhomogeneous renewal equation. We obtain the structure of the solution of this equation in the arithmetic case up to an exponentially small remainder in t.

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Correspondence to M. van den Berg.

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van den Berg, M. Heat equation on the arithmetic von Koch snowflake. Probab Theory Relat Fields 118, 17–36 (2000). https://doi.org/10.1007/s004400000068

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  • DOI: https://doi.org/10.1007/s004400000068

Mathematics Subject Classification (2000)

  • 35K05
  • 60J65
  • 28A80

Key words:

  • Heat equation
  • Arithmetic
  • Snowflake