Annali di Matematica Pura ed Applicata

, Volume 177, Issue 1, pp 37-115

First online:

Nonlinear Hodge theory on manifolds with boundary

  • T. IwaniecAffiliated withDepartment of Mathematics, Syracuse University
  • , C. ScottAffiliated withDepartment of Mathematics, University of Wisconsin
  • , B. StroffoliniAffiliated withDipartimento di Matematica e Applicazioni «R. Caccioppoli», Università

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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.