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Boundary value problems for differential forms on compact Riemannian manifolds. (English) Zbl 1066.31004

One considers \(r\)-forms \(f\) of regularity class \(C^{1,\lambda}(\overline\Omega)\); here \({\mathcal M}\) is a compact, oriented \(n\)-dimensional \(C^\infty\)-Riemannian manifold and \(\Omega\) a compact, oriented \(C^\infty\)-Riemannian submanifold with boundary which consists of a finite number of arcwise connected domains with \(C^\infty\) boundaries and pairwise disjoint closures. As it is well known there exist integral operators \(G\) and \(H\) with \(C^\infty\)-kernels, satisfying \[ \Delta G\phi= \phi- H\phi,\quad\phi\in C^\infty({\mathcal M})^r. \] Using this fact, a generalization of the fundamental theorem for vector fields is derived: For \(f\in C^1(\overline\Omega)^r\), \(0< r< n\), it is shown that \[ f- (H\widetilde f)\lceil\Omega= d\phi+ \delta\theta. \] Here \(d\) is the exterior derivative, \(\delta\) the coderivative, \(\widetilde f\) is the extension of \(f\) one gets by setting \(\widetilde f= 0\) on \(\Omega^c\), \(\phi\) is coclosed and \(\theta\) is closed; an explicit integral representation for the forms \(\phi\) and \(\theta\) is presented. Moreover, solutions of boundary value problems characterizing harmonic fields defined on \(\Omega\) and the corresponding inhomogeneous generalizations are presented.

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31C12 Potential theory on Riemannian manifolds and other spaces
35C15 Integral representations of solutions to PDEs
58A10 Differential forms in global analysis
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