, Volume 177, Issue 1, pp 37-115

Nonlinear Hodge theory on manifolds with boundary

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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, ellipticPDEs, 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 ℒp 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.

Entrata in Redazione il 4 dicembre 1997.
The first two authors were partially supported by NSF grants DMS-9401104 and DMS-9706611. Bianca Stroffolini was supported by CNR. This work started in 1993 when all authors were in Syracuse.