Diploma Thesis


Boundary Value Problems for the de Rham Complex


Let M be a compact oriented Riemannian manifold with boundary. Let η be a differential form on M and χ be a differential form on ∂M. We investigate if there is a differential form ω on M, where its exterior derivative is equal η, and the restriction of ω to ∂M is equal χ. If such an ω exists, we are interested in asking for uniqueness and its properties.

     We have to determine, under which conditions this question makes sense. Considering C we can handle this with standard tools of analysis. But if ω is an L2 form, there may be no way defining its exterior derivative. Restricting ω to the boundary may be impossible as well, as ∂M is an L2 zero set in M.

     It turns out that we may define a restricting operator on the domain of the maximum closed extension of the exterior derivative in L2, mapping differential forms on M to differential forms on ∂M.

     Starting with the smooth case - that means, all differential forms are considered to be C - there arise several difficulties when extending the problem mentioned above to a more general case. Those difficulties are mainly based on the fact that it is not obvious, whether properties of the exterior derivative (e.g. the fact that the exterior derivative and the restriction operator on differential forms to the boundary of M commute) are also valid for its closed extensions, and that some analytic tools can not be used any more when considering manifolds with boundary (e.g. the regularity of an elliptic differential operator).

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