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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
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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 $d$-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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