Some remarks on the characterization of the space of tangential traces of \(H(\text{rot}; \Omega)\) and the construction of an extension operator (Q1911175)

Let \(\Omega\) be a connected, bounded, open subset of \(R^3\), and \(H(\text{rot}; \Omega)\) be a space of the vector functions \(\phi\in (L^2(\Omega))^3\) such that \(\text{rot } \phi\in (L^2(\Omega))^3\). The tangential trace on \(\partial\Omega\) of \(u\in H(\text{rot}; \Omega)\) is \(n\times u_{|\partial\Omega}\) for the unit outward normal vector of \(\partial\Omega\). The authors deal with the space of tangential traces of functions belonging to \(H(\text{rot}; \Omega)\) and construct a linear and continuous extension operator from this trace to \(H(\text{rot}; \Omega)\). They consider also tangential traces on a proper subset of \(\partial\Omega\) and construct corresponding extension operators. Two other results are a solvability theorem for the so-called Dirichlet harmonic vector fields and an existence theorem for a mixed Dirichlet-Neumann boundary value problem.