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Zbl 0963.58003
Iwaniec, T.; Scott, C.; Stroffolini, B.
Nonlinear Hodge theory on manifolds with boundary.
(English)
[J] Ann. Mat. Pura Appl., IV. Ser. 177, 37-115 (1999). ISSN 0373-3114; ISSN 1618-1891/e

Authors' summary: The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, elliptic PDEs called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the ${\cal L}^p$-theory for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated.
[A.P.Stone (Albuquerque)]
MSC 2000:
*58A14 Hodge theory (global analysis)
58J05 Elliptic equations on manifolds, general theory

Keywords: nonlinear Hodge theory; Dirichlet boundary value problems; Hodge systems; Neumann boundary value problems

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