THE CLOSED RANGE THEOREM AND THE TWO-OBSTRUCTIONS CRITERION: APPLICATIONS TO GRAD, DIV, CURL AND THE LAPLACIAN

The criterion of two obstructions and the form-theoretic version of the closed range theorem given in [8] for bounded operators are extended to closed operators between Hilbert spaces. The extension uses the graph norm on the domain and the open mapping theorem on the restriction to the orthogonal complement of the kernel. The resulting theorem applies to the gradient, the divergence, the curl, and the Laplacian, with the Laplacian treated under both Dirichlet and Neumann boundary conditions. For each, the inequality of type (iii) of the theorem is identified with a classical inequality: Poincaré–Wirtinger for the gradient, the Poincaré inequality for the divergence, Friedrichs’ inequalities for the curl on H(curl) and on H0(curl), and the Poincaré inequalities for the Laplacian with Dirichlet and with Neumann conditions. The optimal constant is identified with the smallest positive singular value of the closed operator on the quotient, and in the model cases with the first eigenvalue of the Laplacian or of curl curl. By Banach’s duality, the closed range property and the constant are invariant under passing to the adjoint; this yields the divergence on H0(div), with the Neumann constant µ1, the gradient on H0 1 , with the Dirichlet constant λ1, and the curl on H0(curl), with the same Friedrichs constant, as further instances. The theorem holds for all closed operators with dense domain between Hilbert spaces. An instance where condition (iii) fails, with dense but non-closed range, is also recorded.