The ${L}^{p}$ Neumann problem for the heat equation in non-cylindrical domains
Journées équations aux dérivées partielles (1998), article no. 6, 7 p.

I shall discuss joint work with John L. Lewis on the solvability of boundary value problems for the heat equation in non-cylindrical (i.e., time-varying) domains, whose boundaries are in some sense minimally smooth in both space and time. The emphasis will be on the Neumann problem with data in ${L}^{p}$. A somewhat surprising feature of our results is that, in contrast to the cylindrical case, the optimal results hold when $p=2$, with the situation getting progressively worse as $p$ approaches $1$. In particular, in our setting, the Neumann problem fails to be solvable when the data is taken to belong to the Hardy space ${H}^{1}$.

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title = {The ${L}^p$ {Neumann} problem for the heat equation in non-cylindrical domains},
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publisher = {Universit\'e de Nantes},
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Hofmann, Steve; Lewis, John L. The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains. Journées équations aux dérivées partielles (1998), article  no. 6, 7 p. http://www.numdam.org/item/JEDP_1998____A6_0/

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