## $L^ p$ theory of differential forms on manifolds

HTML articles powered by AMS MathViewer

- by Chad Scott PDF
- Trans. Amer. Math. Soc.
**347**(1995), 2075-2096 Request permission

## 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

- Henri Cartan,
*Differential forms*, Houghton Mifflin Co., Boston, Mass., 1970. Translated from the French. MR**0267477**
J.B. Conway, - P. E. Conner,
*The Neumann’s problem for differential forms on Riemannian manifolds*, Mem. Amer. Math. Soc.**20**(1956), 56. MR**78467** - G. F. D. Duff,
*Differential forms in manifolds with boundary*, Ann. of Math. (2)**56**(1952), 115–127. MR**48136**, DOI 10.2307/1969770 - Bernard Dacorogna,
*Direct methods in the calculus of variations*, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR**990890**, DOI 10.1007/978-3-642-51440-1 - S. K. Donaldson and D. P. Sullivan,
*Quasiconformal $4$-manifolds*, Acta Math.**163**(1989), no. 3-4, 181–252. MR**1032074**, DOI 10.1007/BF02392736 - G. F. D. Duff and D. C. Spencer,
*Harmonic tensors on Riemannian manifolds with boundary*, Ann. of Math. (2)**56**(1952), 128–156. MR**48137**, DOI 10.2307/1969771 - Lawrence C. Evans and Ronald F. Gariepy,
*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660** - Harley Flanders,
*Differential forms with applications to the physical sciences*, 2nd ed., Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1989. MR**1034244** - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0
Piotr Hajlasz, - Tadeusz Iwaniec,
*$p$-harmonic tensors and quasiregular mappings*, Ann. of Math. (2)**136**(1992), no. 3, 589–624. MR**1189867**, DOI 10.2307/2946602 - Tadeusz Iwaniec and Adam Lutoborski,
*Integral estimates for null Lagrangians*, Arch. Rational Mech. Anal.**125**(1993), no. 1, 25–79. MR**1241286**, DOI 10.1007/BF00411477 - Tadeusz Iwaniec and Gaven Martin,
*Quasiregular mappings in even dimensions*, Acta Math.**170**(1993), no. 1, 29–81. MR**1208562**, DOI 10.1007/BF02392454 - Kunihiko Kodaira,
*Harmonic fields in Riemannian manifolds (generalized potential theory)*, Ann. of Math. (2)**50**(1949), 587–665. MR**31148**, DOI 10.2307/1969552 - H. Blaine Lawson Jr. and Marie-Louise Michelsohn,
*Spin geometry*, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR**1031992** - Charles B. Morrey Jr.,
*Multiple integrals in the calculus of variations*, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR**0202511**, DOI 10.1007/978-3-540-69952-1 - Marston Morse,
*Global variational analysis*, Mathematical Notes, No. 16, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1976. Weierstrass integrals on a Riemannian manifold. MR**0494242**
Sunil Mukhi and N. Mukunda, - Raghavan Narasimhan,
*Analysis on real and complex manifolds*, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR**0251745** - Joel W. Robbin, Robert C. Rogers, and Blake Temple,
*On weak continuity and the Hodge decomposition*, Trans. Amer. Math. Soc.**303**(1987), no. 2, 609–618. MR**902788**, DOI 10.1090/S0002-9947-1987-0902788-8 - H. L. Royden,
*Real analysis*, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR**0151555** - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095**
C.H. Scott, ${L^p}$ - L. M. Sibner and R. J. Sibner,
*A non-linear Hodge-de-Rham theorem*, Acta Math.**125**(1970), 57–73. MR**281231**, DOI 10.1007/BF02392330 - K. Uhlenbeck,
*Regularity for a class of non-linear elliptic systems*, Acta Math.**138**(1977), no. 3-4, 219–240. MR**474389**, DOI 10.1007/BF02392316 - Frank W. Warner,
*Foundations of differentiable manifolds and Lie groups*, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR**722297**, DOI 10.1007/978-1-4757-1799-0
Eberhard Zeidler,

*A course in functional analysis*(2nd ed.), Macmillan, New York, 1988.

*Note on Meyers-Serrin’s Theorem*, Expositiones Math. (to appear).

*Introduction to topology, differential geometry and group theory for physicists*, Wiley Eastern Limited, New Delhi, 1990.

*theory of differential forms on manifolds*, Ph.D. Thesis, Mathematics Department, Syracuse University, June, 1993.

*Nonlinear functional analysis and its applications*, Springer-Verlag, New York, 1990.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**347**(1995), 2075-2096 - MSC: Primary 58A14; Secondary 58G03
- DOI: https://doi.org/10.1090/S0002-9947-1995-1297538-7
- MathSciNet review: 1297538