Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on boundary value problems for the heat equation in Lipschitz cylinders


Authors: Russell M. Brown and Zhong Wei Shen
Journal: Proc. Amer. Math. Soc. 119 (1993), 585-594
MSC: Primary 35K05; Secondary 35A20, 35D05, 35K20
DOI: https://doi.org/10.1090/S0002-9939-1993-1156466-1
MathSciNet review: 1156466
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the initial Dirichlet problem and the initial Neumann problem for the heat equation in Lipschitz cylinders, with boundary data in mixed norm spaces $ {L^q}(0,T,{L^p}(\partial \Omega ))$.


References [Enhancements On Off] (What's this?)

  • [B1] R. M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 330-379. MR 987761 (90d:35118)
  • [B2] -, The initial-Neumann problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 320 (1990), 1-52. MR 1000330 (90k:35112)
  • [BS1] R. M. Brown and Z. Shen, The initial-Dirichlet problem for a fourth-order parabolic equation in Lipschitz cylinders, Indiana Univ. J. Math. 39 (1990), 1313-1353. MR 1087194 (92b:35075)
  • [BS2] -, Boundary value problems in Lipschitz cylinders for three-dimensional parabolic systems, Revista Mat. Iberó. 8 (1992), 271-303. MR 1202412 (94b:35133)
  • [CMM] R. R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $ {L^2}$ pour les courbes Lipschitziennes, Ann. of Math. (2) 116 (1982), 361-387. MR 672839 (84m:42027)
  • [DK] B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the Neumann problem in $ {L^p}$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), 437-465. MR 890159 (88d:35044)
  • [FR] E. B. Fabes and N. M. Rivière, Dirichlet and Neumann problems for the heat equation in $ {C^1}$-cylinders, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, RI, 1979, pp. 179-196.
  • [FGS] E. B. Fabes, N. Garofalo, and S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), 536-565. MR 857210 (88d:35089)
  • [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 709573 (85c:35034)
  • [FSt] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • [JK] D. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 703-707. MR 598688 (84a:35064)
  • [K] C. E. Kenig, Elliptic boundary value problems on Lipschitz domains, Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 131-183. MR 864372 (88a:35066)
  • [NSt] A. Nagel and E. M. Stein, Lectures on pseudo-differential operators, Princeton Univ. Press, Princeton, NJ, 1979. MR 549321 (82f:47059)
  • [S] Z. Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, Amer. J. Math. 113 (1991), 293-373. MR 1099449 (92a:35133)
  • [St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [V] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. MR 769382 (86e:35038)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35K05, 35A20, 35D05, 35K20

Retrieve articles in all journals with MSC: 35K05, 35A20, 35D05, 35K20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1156466-1
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society