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

### Luigi Fontana, Steven G. Krantz, and Marco M. Peloso

**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.

*Copyright American Mathematical Society 1996*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 103-107
- Publisher Identifier: S1079-6762(95)03002-2
- 1991
*Mathematics Subject Classification*. Primary 35J55; Secondary 35S15, 35N15, 58A14, 58G05.
*Key words and phrases*. Hodge theory,
de Rham complex, "$\overline{\partial}$"-Neumann complex,
elliptic estimates,
subelliptic estimates,
pseudodifferential boundary problems
- Received by the editors July 29, 1995
- Communicated by Robert Lazarsfeld
- Comments

**Luigi Fontana**

Dipartimento di Matematica

Via Saldini 50

Università di Milano

20133 Milano (Italy)

*E-mail address:* `fontana@vmimat.mat.unimi.it`

**Steven G. Krantz**

Department of Mathematics

Washington University

St.
Louis, MO 63130
(U.S.A)

*E-mail address:* `sk@math.wustl.edu`

**Marco M. Peloso**

Dipartimento di Matematica

Politecnico di Torino

10129
Torino (Italy)

*E-mail address:* `peloso@polito.it`

Second author supported in part by the National Science Foundation

Third author supported in part by the Consiglio Nazionale delle Ricerche

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