The L p Neumann problem for the heat equation in non-cylindrical domains

Steve Hofmann; John L. Lewis

Journées équations aux dérivées partielles (1998)

  • page 1-7
  • ISSN: 0752-0360

Abstract

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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 .

How to cite

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Hofmann, Steve, and Lewis, John L.. "The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains." Journées équations aux dérivées partielles (1998): 1-7. <http://eudml.org/doc/93363>.

@article{Hofmann1998,
abstract = {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 $H^1$.},
author = {Hofmann, Steve, Lewis, John L.},
journal = {Journées équations aux dérivées partielles},
keywords = {minimally smooth boundary; Neumann problem with data in ; Hardy space},
language = {eng},
pages = {1-7},
publisher = {Université de Nantes},
title = {The $\{L\}^p$ Neumann problem for the heat equation in non-cylindrical domains},
url = {http://eudml.org/doc/93363},
year = {1998},
}

TY - JOUR
AU - Hofmann, Steve
AU - Lewis, John L.
TI - The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains
JO - Journées équations aux dérivées partielles
PY - 1998
PB - Université de Nantes
SP - 1
EP - 7
AB - 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 $H^1$.
LA - eng
KW - minimally smooth boundary; Neumann problem with data in ; Hardy space
UR - http://eudml.org/doc/93363
ER -

References

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  9. [HL] S. Hofmann and J.L. Lewis, L2 solvability and representiation by caloric layer potentials in time-varying domains, Ann. of Math, 144 (1996), 349-420. Zbl0867.35037MR97h:35072
  10. [K] C. Kenig, Elliptic boundary value problems on Lipschitz domains, in Beijing Lecutres in harmonic Analysis, E.M. Stein, ed., Ann. of Math Studies 112 (1986), 131-183. Zbl0624.35029MR88a:35066
  11. [LM] J.L. Lewis and M.A.M. Murray, The method of layer potentials for the heat equation in time varying domains, Memoirs A.M.S. Vol. 114, Number 545, 1995. Zbl0826.35041MR96e:35059
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  13. [V] C. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. of Functional Analysis 59 (1984), 572-611. Zbl0589.31005MR86e:35038

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