The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains

Hofmann, Steve; Lewis, John L.

The

L^{p}

Neumann problem for the heat equation in non-cylindrical domains

Hofmann, Steve ; Lewis, John L.

Journées équations aux dérivées partielles (1998), article no. 6, 7 p.

Résumé

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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     author = {Hofmann, Steve and Lewis, John L.},
     title = {The ${L}^p$ {Neumann} problem for the heat equation in non-cylindrical domains},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     pages = {1--7},
     publisher = {Universit\'e de Nantes},
     year = {1998},
     mrnumber = {1640379},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_1998____A6_0/}
}

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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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