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It is well known that, with a particular choice of norm, the classicaldouble-layer potential operator $D$ has essential norm $<1/2$ as an operator onthe natural trace space $H^{1/2}(\Gamma)$ whenever $\Gamma$ is the boundary ofa bounded Lipschitz domain. This implies, for the standard second-kind boundaryintegral equations for the interior and exterior Dirichlet and Neumann problemsin potential theory, convergence of the Galerkin method in $H^{1/2}(\Gamma)$for any sequence of finite-dimensional subspaces $(\mathcal{H}_N)_{N=1}^\infty$that is asymptotically dense in $H^{1/2}(\Gamma)$. Long-standing open questionsare 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 Lipschitzpolyhedra in 3-d, the weaker condition holds that the operators $\pm\frac{1}{2}I+D$ are compact perturbations of coercive operators -- this anecessary and sufficient condition for the convergence of the Galerkin methodfor every sequence of subspaces $(\mathcal{H}_N)_{N=1}^\infty$ that isasymptotically dense in $L^2(\Gamma)$. We settle these open questionsnegatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitzconstant equal to one for which the essential norm of $D$ is $\geq 1/2$, andexamples 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 ofLipschitz 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 realor complex $\lambda$ with $|\lambda|\leq C$. Finally, we resolve negatively arelated open question in the convergence theory for collocation methods.

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