The initial-Neumann problem for the heat equation in Lipschitz cylinders

Brown, Russell M.

doi:10.1090/S0002-9947-1990-1000330-7

The initial-Neumann problem for the heat equation in Lipschitz cylinders
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Trans. Amer. Math. Soc. 320 (1990), 1-52 Request permission

Abstract:

We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in ${L^p}$, $1 < p < 2+\varepsilon$, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given 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 half of a time derivative in ${L^p}$, $1 < p < 2+\varepsilon$, 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 ${L^q}$, $2 - \varepsilon ’ < q < \infty$ and BMO may be represented as double-layer heat potentials.

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