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

Abstract

AbstractIt is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm $$<1/2$$ < 1 / 2 as an operator on the natural trace space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) whenever $$\Gamma $$ 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 $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) for any sequence of finite-dimensional subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . Long-standing open questions are whether the essential norm is also $$<1/2$$ < 1 / 2 for D as an operator on $$L^2(\Gamma )$$ L 2 ( Γ ) for all Lipschitz $$\Gamma $$ Γ in 2-d; or whether, for all Lipschitz $$\Gamma $$ 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 $$\pm \frac{1}{2}I+D$$ ± 1 2 I + 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 $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$L^2(\Gamma )$$ 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 $$\ge 1/2$$ ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators $$\pm \frac{1}{2}I +D$$ ± 1 2 I + D are not coercive plus compact. We also give, for every $$C>0$$ C > 0 , examples of Lipschitz polyhedra for which the essential norm is $$\ge C$$ ≥ C and for which $$\lambda I+D$$ λ I + D is not a compact perturbation of a coercive operator for any real or complex $$\lambda $$ λ with $$|\lambda |\le C$$ | 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 $$L^2(\Gamma )$$ 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 $$C(\Gamma )$$ C ( Γ ) , equivalent to the standard supremum norm, for which the essential norm of D on $$C(\Gamma )$$ C ( Γ ) is $$<1/2$$ < 1 / 2 .