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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The oblique derivative problem for the heat equation in Lipschitz cylinders


Author: Russell M. Brown
Journal: Proc. Amer. Math. Soc. 107 (1989), 237-250
MSC: Primary 35K05
MathSciNet review: 987608
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Abstract: We consider a class of initial-boundary value problems for the heat equation on $ (0.T) \times \Omega $ with $ \Omega $ a bounded Lipschitz domain in $ {{\mathbf{R}}^n}$. On the lateral boundary, $ (0,T) \times \partial \Omega = {\Sigma _T}$, we specify $ \left\langle {\alpha ,\nabla u} \right\rangle $ where $ \nabla u$ denotes the spatial gradient of the solution and $ \alpha :{\Sigma _T} \to \{ x:\vert x\vert = 1\} $ is a continuous vector field satisfying $ \left\langle {\alpha ,\nu } \right\rangle \geq \mu > 0$ with $ \nu $ the unit normal to $ \partial \Omega $. On the initial surface, $ \{ 0\} \times \Omega $, we require that the solution vanish. The lateral data is taken from $ {L^p}({\Sigma _T})$. For $ p \in (2 - ,\infty )$, we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0987608-5
PII: S 0002-9939(1989)0987608-5
Keywords: Heat equation, initial-boundary value problems, nonsmooth domains
Article copyright: © Copyright 1989 American Mathematical Society