Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space
Authors:
Luigi Fontana, Steven G. Krantz and Marco M. Peloso
Journal:
Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 103-107
MSC (1991):
Primary 35J55, 35S15, 35N15, 58A14, 58G05
DOI:
https://doi.org/10.1090/S1079-6762-95-03002-2
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 $\bar {\partial }$-Neumann problem in the Sobolev topology is studied. A new $\bar {\partial }$-Neumann boundary condition is obtained, and the corresponding subelliptic estimate derived.
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- Louis Boutet de Monvel, Comportement d’un opérateur pseudo-différentiel sur une variété à bord. II. Pseudo-noyaux de Poisson, J. Analyse Math. 17 (1966), 255–304 (French). MR 239255, DOI https://doi.org/10.1007/BF02788661
- Louis Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), no. 1-2, 11–51. MR 407904, DOI https://doi.org/10.1007/BF02392024
- G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. MR 0461588
- L. Fontana, S. G. Krantz, M. M. Peloso, Hodge theory in the Sobolev topology for the de Rham complex, preprint.
- L. Fontana, S. G. Krantz, M. M. Peloso, The $\bar {\partial }$-Neumann problem in the Sobolev topology, in progress.
- J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 153030, DOI https://doi.org/10.2307/1970506
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- L. Boutet de Monvel, Comportement d’un opérator pseudo-différential sur une variété à bord. I. Journal d’Anal. Math. 17(1966), 241-253.
- L. Boutet de Monvel, ibid, 254-304.
- L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126(1971), 11-51.
- G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, 1972.
- L. Fontana, S. G. Krantz, M. M. Peloso, Hodge theory in the Sobolev topology for the de Rham complex, preprint.
- L. Fontana, S. G. Krantz, M. M. Peloso, The $\bar {\partial }$-Neumann problem in the Sobolev topology, in progress.
- J. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, Ann. Math. 78(1963), 112-148; II, ibid. 79(1964), 450-472. ; MR 34:8010
- D. H. Phong, thesis, Princeton University, 1977.
- R. Sweeney, The $d$-Neumann problem, Acta Math. 120(1968), 224-277.
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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
Keywords:
Hodge theory,
de Rham complex,
$\bar {\partial }$-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
Article copyright:
© Copyright 1996
American Mathematical Society
