A Neumann problem for harmonic forms in \(C^1\)-domains (Q5945664)

Let \(\Omega\) be a bounded domain in \(\mathbb R^n\), \(n\geq 3\), such that \(\partial\Omega\) is of class \(C^1\). The authors look at solutions of the Neumann problem in \(\Omega\) in the space of differential forms whose coefficients and its first order derivatives have interior nontangential trace in \(L^p(\partial\Omega)\). First it is shown that the Neumann problem may be reduced to a singular integral equation on \(\partial\Omega\). Since this integral equation turns out to be solvable for suitable boundary data, an existence theorem for the corresponding Neumann problem follows. In a subsequent section it is proved that the solutions of the homogeneous problem are closed forms which implies a uniqueness theorem for the original problem.