The initial-neumann problem for the heat equation in lipschitz cylinders

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We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateraldata is in Lp, 1 < p < 2 + ε, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates aregiven for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space H1. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and haif of a time derivative in Lp, 1 < p < 2 + ε, with a corresponding Hardy space result when p = 1. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in Lq, 2 − ε < q < ∞ and BMO may be represented as double-layer heat potentials.

Original languageEnglish
Pages (from-to)1-52
Number of pages52
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - Jul 1990


  • Heatequation
  • Initial-boundary value problems
  • Non smooth domains

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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