Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The initial-Neumann problem for the heat equation in Lipschitz cylinders


Author: Russell M. Brown
Journal: Trans. Amer. Math. Soc. 320 (1990), 1-52
MSC: Primary 35K05; Secondary 31B35, 35C15, 42B30, 46E35
DOI: https://doi.org/10.1090/S0002-9947-1990-1000330-7
MathSciNet review: 1000330
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in $ {L^p}$, $ 1 < p < 2+\varepsilon $, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space $ {H^1}$. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in $ {L^p}$, $ 1 < p < 2+\varepsilon $, with a corresponding Hardy space result when $ p = 1$. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in $ {L^q}$, $ 2 - \varepsilon ' < q < \infty $ and BMO may be represented as double-layer heat potentials.


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

  • [A1] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890-896. MR 0217444 (36:534)
  • [A2] -, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa 22 (1968), 607-694. MR 0435594 (55:8553)
  • [B1] R. M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 339-379. MR 987761 (90d:35118)
  • [B2] R. M. Brown, Area integral estimates for caloric functions, Trans. Amer. Math. Soc. 315 (1989), 565-589. MR 994163 (90j:35103)
  • [C1] A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324-1327. MR 0466568 (57:6445)
  • [C2] -, Boundary value problems in Lipschitzian domains, Recent Progress in Fourier Analysis, Elsevier Science Publishers, 1985, pp. 33-48.
  • [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. 116 (1982), 361-387. MR 672839 (84m:42027)
  • [CW] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1976), 569-645. MR 0447954 (56:6264)
  • [DK] B. E. J. Dahlberg and C. E. Kenig, Hardy space and the Neumann problem in $ {L^p}$ for Laplace's equation in Lipschitz domains, Ann. of Math. 125 (1987), 437-466. MR 890159 (88d:35044)
  • [FJ] E. B. Fabes and M. Jodeit, Jr., $ {L^p}$-boundary value problems for parabolic equations, Bull. Amer. Math. Soc. 74 (1968), 1098-1102. MR 0233081 (38:1404)
  • [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, R.I., 1979, pp. 179-196. MR 545307 (81b:35044)
  • [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] E. B. Fabes and D. Stroock, The $ {L^p}$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), 997-1016. MR 771392 (86g:35057)
  • [GJ] J. B. Garnett and P. W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), 351-371. MR 658065 (85d:42021)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, 1983. MR 737190 (86c:35035)
  • [GNR] A. Grimaldi-Piro, U. Neri and F. Ragnedda, Invertibility of heat potentials in BMO norms, Rend. Sem. Mat. Univ. Padova 75 (1986), 77-90. MR 847659 (87k:35118)
  • [K] J. Kemper, Temperatures in several variables: Kernel functions, representations and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243-262. MR 0294903 (45:3971)
  • [Ko] R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78 (1982), 131-172. MR 648942 (83f:73018)
  • [LSU] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and quasilinear equations of parabolic type, Transl. Math. Mono., vol. 23, Amer. Math. Soc., Providence, R.I., 1968.
  • [M] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134, Correction 20 (1967), 231-236.. MR 0159139 (28:2357)
  • [St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton N.J., 1970. MR 0290095 (44:7280)
  • [Va] N. Varopoulos, BMO functions and the $ \bar \partial $ equation, Pacific J. Math. 71 (1977), 221-273. MR 0508035 (58:22639a)
  • [V] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation on Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. MR 769382 (86e:35038)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K05, 31B35, 35C15, 42B30, 46E35

Retrieve articles in all journals with MSC: 35K05, 31B35, 35C15, 42B30, 46E35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1000330-7
Keywords: Heat equation, initial-boundary value problems, nonsmooth domains
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society