Hodge decomposition. A method for solving boundary value problems
Recall that a manifold with boundary \(M\) is called a \(\partial\)-manifold if \(M\) is equipped with a smooth atlas \({\mathcal A}\) such that \({\mathcal A}\) is oriented and \(M\) is complete as a metric space. (The usual definition does not require completeness, but the author requires it in view of his study of Sobolev spaces.) The author proves a generalization of the Hodge decomposition theorem (called the Hodge-Morrey decomposition) for differential forms on a \(\partial\)-manifold \(M\) of Sobolev class \(W^{s,p}\). His proof uses a regularity result for Dirichlet (and Neumann) potentials of a given differential form, showing that these potentials are weak solutions of an elliptic boundary value problem. He also obtains a generalization of the Friedrichs decomposition of the space of harmonic fields to this case. These decompositions are applied to the study of various boundary value problems for differential forms, in particular to a mixed inhomogeneous problem for the exterior derivative i.e., the author constructs a solution to the equation \(d \omega = \chi\) with the full boundary value \(\omega |_{\partial M}\) prescribed \((\omega\) is a \(k\)-form on a \(\partial\)-manifold \(M)\). The construction is based on an extension problem for a differential form \(\eta\) with \(\eta|_{\partial M}\) prescribed together with the normal exterior derivatives of \(\eta\) and its tangential codifferential. In an appendix to Chapter 2, Jan Wenzelburger proves that decompositions of a Hilbert space \(H\) depend smoothly on a given inner product belonging to the Banach subspace of inner products on \(H\) in the space of all continuous bilinear forms on \(H\). This result is applied to show that the Hodge-Morrey, Friedrichs, and Helmholtz decompositions depend smoothly on the Riemannian metric.
scientific article; zbMATH DE number 783743