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 .

@incollection{JEDP_1998____A6_0,
     author = {Steve Hofmann and John L. Lewis},
     title = {The ${L}^p$ {Neumann} problem for the heat equation in non-cylindrical domains},
     booktitle = {},
     series = {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 = {https://proceedings.centre-mersenne.org/item/JEDP_1998____A6_0/}
}
TY  - JOUR
AU  - Steve Hofmann
AU  - John L. Lewis
TI  - The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains
JO  - Journées équations aux dérivées partielles
PY  - 1998
SP  - 1
EP  - 7
PB  - Université de Nantes
UR  - https://proceedings.centre-mersenne.org/item/JEDP_1998____A6_0/
LA  - en
ID  - JEDP_1998____A6_0
ER  - 
%0 Journal Article
%A Steve Hofmann
%A John L. Lewis
%T The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains
%J Journées équations aux dérivées partielles
%D 1998
%P 1-7
%I Université de Nantes
%U https://proceedings.centre-mersenne.org/item/JEDP_1998____A6_0/
%G en
%F JEDP_1998____A6_0
Steve Hofmann; John L. Lewis. The ${L}^p$ Neumann problem for the heat equation in non-cylindrical domains. Journées équations aux dérivées partielles (1998), article  no. 6, 7 p. https://proceedings.centre-mersenne.org/item/JEDP_1998____A6_0/

[Br1] R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 359-379. | MR | Zbl

[Br2] R. Brown, The initial-Neumann problem for the heat equation in Lipschitz cylinders Trans. A.M.S. 320 (1990), 1-52. | MR | Zbl

[D] B. Dahlberg, Poisson semigroups and singular integrals, Proc. A.M.S. 97 (1986), 41-48. | MR | Zbl

[DK] B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in Lp for Laplace's equation in Lipschitz domains, Ann. of Math. 125 (1987), 437-466. | MR | Zbl

[DKPV] B. Dahlberg, C. Kenig, J. Pipher and G. Verchota, Area integral estimates and maximum principles for higher order elliptic equations and systems on Lipschitz domains, to appear. | Numdam | Zbl

[FR] E.B. Fabes and N. Riviere, Symbolic calculus of kernels with mixed homogeneity, in Singular Integrals, A.P. Calderón, ed., Proc. Symp. Pure Math., Vol. 10, A.M.S. Providence, 1967, pp. 106-127. | MR | Zbl

[FS] E.B. Fabes and S. Salsa, Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 279 (1983), 635-650. | MR | Zbl

[H] S. Hofmann, A characterization of commutators of parabolic singular inegrals, in Fourier Analysis and Parital Differential Equations, J. García-Cuerva, E. Hernández, F. Soria, and J.-L. Torrea, eds., Studies in Advanced Mathematics, CRC press, Boca Raton, 1995, pp. 195-210. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[Stz] R. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana U. Math. J. 29 (1980), 539-558. | MR | Zbl

[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. | MR | Zbl