zbMATH — the first resource for mathematics

Compactness theorems for differential forms. (English) Zbl 0649.58002
This paper proves that the set of forms \(\omega\) on a Riemannian manifold such that \(\| \omega \| _{L\quad p}+\| d\omega \| _{L\quad p}\leq 1,\) \((1<p<\infty)\), although not compact in L p in general, is compact for an L p analogue of the Whitney flat norm. The corresponding result with L p replaced by the space of currents with finite mass is also proved. Both follow from the Hodge decomposition and elliptic estimates. Special cases yield very simple proofs of the Federer-Fleming compactness theorem for normal currents and of an L p- version of the div-curl lemma (“compensated compactness”). The relation of the new flat norm to the L p-norm is investigated, as well as its behavior under smoothing by the heat kernel. In this connection, an example is given of a mass minimizing current which is not “smooth” (i.e., not supported by a submanifold).
Reviewer: S.Kichenassamy

58A10 Differential forms in global analysis
58J10 Differential complexes
49Q20 Variational problems in a geometric measure-theoretic setting
Full Text: DOI
[1] Geometric Measure Theory, Springer, Berlin, Heidelberg, New York, 1969.
[2] Murat, Ann. Scuola Norm. Sup. Pisa 5 pp 489– (1978)
[3] Ann. Scuola Norm. Sup. Pisa 8 pp 64– (1981)
[4] See also Compensated compactness and applications to p.d.e., in Nonlinear analysis and mechanics, Meriot-Watt Symposium, Vol. IV, Ed., Res. Notes in Math. 39, Pitman, 1979, pp. 136–212.
[5] , and , Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
[6] Sarason, Ann. Mat. Pura Appl. 92 pp 23– (1972)
[7] Kato, Indiana Univ. Math. J. 24 pp 979– (1975)
[8] Schulenberger, Ann. Mat. Pura Appl. 88 pp 229– (1971)
[9] Ann. Mat. Pura Appl. 92 pp 77– (1972)
[10] and , Mathematical Theory of Elastic and Elastoplastic Bodies, Elsevier, 1981.
[11] Kichenassamy, Man. Math. 5 pp 281– (1987)
[12] de Figueiredo, Comm. Pure Appl. Math. 16 pp 63– (1963)
[13] Federer, Ann. of Math. 72 pp 458– (1960)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.