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ISSN 1079-6762

Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space

Author(s): Luigi Fontana; Steven G. Krantz; Marco M. Peloso
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 103-107.
MSC (1991): Primary 35J55, 35S15, 35N15, 58A14, 58G05
Comment(s): Additional information about this paper
MathSciNet review: 1369640
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Abstract: The Hodge theory of the de Rham complex in the setting of the Sobolev topology is studied. As a result, a new elliptic boundary value problem is obtained. Next, the Hodge theory of the $\overline{\partial}$ -Neumann problem in the Sobolev topology is studied. A new $\overline{\partial}$ -Neumann boundary condition is obtained, and the corresponding subelliptic estimate derived.

References:

BDM1: L. Boutet de Monvel, Comportement d'un opérator pseudo-différential sur une variété à bord. I. Journal d'Anal. Math. 17(1966), 241-253. MR 39:611
BDM2: L. Boutet de Monvel, ibid, 254-304. MR 39:612
BDM3: L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126(1971), 11-51. MR 53:11674
FOK: G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, 1972. MR 57:1573
FKP1: L. Fontana, S. G. Krantz, M. M. Peloso, Hodge theory in the Sobolev topology for the de Rham complex, preprint.
FKP2: L. Fontana, S. G. Krantz, M. M. Peloso, The $\overline{\partial}$ -Neumann problem in the Sobolev topology, in progress.
KOH: J. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, Ann. Math. 78(1963), 112-148; II, ibid. 79(1964), 450-472. MR 27:2999; MR 34:8010
PHO: D. H. Phong, thesis, Princeton University, 1977.
SWE: R. Sweeney, The -Neumann problem, Acta Math. 120(1968), 224-277. MR 37:2250

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Additional Information:

Luigi Fontana
Affiliation: Dipartimento di Matematica Via Saldini 50 Università di Milano 20133 Milano (Italy)
Email: fontana@vmimat.mat.unimi.it

Steven G. Krantz
Affiliation: Department of Mathematics Washington University St. Louis, MO 63130 (U.S.A.)
Email: sk@math.wustl.edu

Marco M. Peloso
Affiliation: Dipartimento di Matematica Politecnico di Torino 10129 Torino (Italy)
Email: peloso@polito.it

DOI: 10.1090/S1079-6762-95-03002-2
PII: S 1079-6762(95)03002-2
Keywords: Hodge theory, de Rham complex, $\dbar$-Neumann complex, elliptic estimates, subelliptic estimates, pseudodifferential boundary problems
Received by editor(s): July 29, 1995
Additional Notes: Second author supported in part by the National Science Foundation
Third author supported in part by the Consiglio Nazionale delle Ricerche
Communicated by: Robert Lazarsfeld
Copyright of article: Copyright 1996, American Mathematical Society

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