Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm < 1 / 2 as an operator on the natural trace space H ^{1 / 2}(Γ) whenever Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H ^{1 / 2}(Γ) for any sequence of finite-dimensional subspaces (HN)N=1∞ that is asymptotically dense in H ^{1 / 2}(Γ). Long-standing open questions are whether the essential norm is also < 1 / 2 for D as an operator on L ²(Γ) for all Lipschitz Γ in 2-d; or whether, for all Lipschitz Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators ±12I+D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces (HN)N=1∞ that is asymptotically dense in L ²(Γ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators ±12I+D are not coercive plus compact. We also give, for every C> 0 , examples of Lipschitz polyhedra for which the essential norm is ≥ C and for which λI+ D is not a compact perturbation of a coercive operator for any real or complex λ with | λ| ≤ C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L ²(Γ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Γ) , equivalent to the standard supremum norm, for which the essential norm of D on C(Γ) is < 1 / 2.