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)



$ L\sp p$ theory of differential forms on manifolds

Author: Chad Scott
Journal: Trans. Amer. Math. Soc. 347 (1995), 2075-2096
MSC: Primary 58A14; Secondary 58G03
MathSciNet review: 1297538
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we establish a Hodge-type decomposition for the $ {L^p}$ space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an $ {L^p}$ estimate which contains, as a special case, the $ {L^2}$ result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the $ {L^p}$ boundedness of Green's operator which we use in developing the $ {L^p}$ theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the $ {L^p}$ norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the $ A$-harmonic equation.

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

  • [C] H. Cartan, Differential forms, Houghton Mifflin, Boston, MA, 1970. MR 0267477 (42:2379)
  • [Co] J.B. Conway, A course in functional analysis (2nd ed.), Macmillan, New York, 1988.
  • [Con] P.E. Conner, The Neumann's problem for differential forms on Riemannian manifolds, Mem. Amer. Math. Soc. 20 (1956). MR 0078467 (17:1197e)
  • [D] G.F.D. Duff, Differential forms on manifolds with boundary, Ann. of Math. 56 (1952), 115-127. MR 0048136 (13:986e)
  • [Da] B. Dacorongna, Direct methods in the calculus of variations, Springer-Verlag, New York, 1989. MR 990890 (90e:49001)
  • [DS] S.K. Donaldson and D.P. Sullivan, Quasiconformal $ 4$-manifolds, Acta Math. 163 (1989), 181-252. MR 1032074 (91d:57012)
  • [DSp] G.F.D. Duff and D.C. Spencer, Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math. 56 (1952), 128-156. MR 0048137 (13:987a)
  • [EG] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
  • [F] Harley Flanders, Differential forms with applications to the physical sciences, Dover, New York, 1989. MR 1034244 (90k:53001)
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order (2nd ed.), Springer-Verlag, New York, 1983. MR 737190 (86c:35035)
  • [H] Piotr Hajlasz, Note on Meyers-Serrin's Theorem, Expositiones Math. (to appear).
  • [I] T. Iwaniec, $ p$-harmonic tensors and quasiregular mappings, Ann. of Math. 136 (1992), 589-624. MR 1189867 (94d:30034)
  • [IL] T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. (to appear). MR 1241286 (95c:58054)
  • [IM] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math.170 (1993), 29-81. MR 1208562 (94m:30046)
  • [K] K. Kodaira, Harmonic fields in Riemannian manifolds, Ann. of Math. 50 (1949), 587-665. MR 0031148 (11:108e)
  • [LM] H.B. Lawson and M.L. Michelsohn, Spin geometry, Princeton Univ. Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
  • [M] C.B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. MR 0202511 (34:2380)
  • [Mo] Marston Morse, Global variational analysis, Weierstrass integrals on a Riemannian manifold, Princeton Univ. Press, Princeton, NJ, 1976. MR 0494242 (58:13150)
  • [MM] Sunil Mukhi and N. Mukunda, Introduction to topology, differential geometry and group theory for physicists, Wiley Eastern Limited, New Delhi, 1990.
  • [N] R. Narasimhan, Analysis on real and complex manifolds, Elsevier Science B.V., Amsterdam, 1968. MR 0251745 (40:4972)
  • [RRT] J.W. Robbin, R.C. Rogers and B. Temple, On weak continuity and the Hodge decomposition, Trans. Amer. Math. Soc. 303 (1987), 609-618. MR 902788 (88m:58004)
  • [R] H.L. Royden, Real analysis, (3rd ed.), Macmillan, New York, 1988. MR 0151555 (27:1540)
  • [S] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [Sc] C.H. Scott, $ {L^p}$ theory of differential forms on manifolds, Ph.D. Thesis, Mathematics Department, Syracuse University, June, 1993.
  • [SS] L.M. Sibner and R.B. Sibner, A non-linear Hodge De Rham Theorem, Acta Math. 125 (1970), 57-73. MR 0281231 (43:6950)
  • [U] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219-250. MR 0474389 (57:14031)
  • [W] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, New York, 1983. MR 722297 (84k:58001)
  • [Z] Eberhard Zeidler, Nonlinear functional analysis and its applications, Springer-Verlag, New York, 1990.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A14, 58G03

Retrieve articles in all journals with MSC: 58A14, 58G03

Additional Information

Keywords: Differential form, Hodge decomposition, harmonic integral, Sobolev space
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society