## Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains
Chandler-Wilde, S. N. and Spence, E. A.
(2021)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1007/s00211-021-01256-x ## Abstract/SummaryIt 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 $H^{1/2}(\Gamma)$ whenever $\Gamma$ 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 $H^{1/2}(\Gamma)$ for any sequence of finite-dimensional subspaces $(\cH_N)_{N=1}^\infty$ that is asymptotically dense in $H^{1/2}(\Gamma)$. Long-standing open questions are whether the essential norm is also $<1/2$ for $D$ as an operator on $L^2(\Gamma)$ for all Lipschitz $\Gamma$ in 2-d; or whether, for all Lipschitz $\Gamma$ 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 $\pm \frac{1}{2}I+D$ are compact perturbations of coercive operators -- this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $(\cH_N)_{N=1}^\infty$ that is asymptotically dense in $L^2(\Gamma)$. 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 $\geq 1/2$, and examples with Lipschitz constant two for which the operators $\pm \frac{1}{2}I +D$ are not coercive plus compact. We also give, for every $C>0$, examples of Lipschitz polyhedra for which the essential norm is $\geq C$ and for which $\lambda I+D$ is not a compact perturbation of a coercive operator for any real or complex $\lambda$ with $|\lambda|\leq C$. 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 $L^2(\Gamma)$ 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 $C(\Gamma)$, equivalent to the standard supremum norm, for which the essential norm of $D$ on $C(\Gamma)$ is $<1/2$.
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