## Abstract

We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateraldata is in L^{p}, 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 H^{1}. 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 L^{p}, 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 L^{q}, 2 − ε^{′} < q < ∞ and BMO may be represented as double-layer heat potentials.

Original language | English |
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Pages (from-to) | 1-52 |

Number of pages | 52 |

Journal | Transactions of the American Mathematical Society |

Volume | 320 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1990 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics