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Math."],"published-print":{"date-parts":[[2022,2]]},"abstract":"Abstract<\/jats:title>It is well known that, with a particular choice of norm, the classical double-layer potential operatorD<\/jats:italic>has essential norm$$<1\/2$$<\/jats:tex-math><<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>as an operator on the natural trace space$$H^{1\/2}(\\Gamma )$$<\/jats:tex-math>H<\/mml:mi>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>whenever$$\\Gamma $$<\/jats:tex-math>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>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$$H^{1\/2}(\\Gamma )$$<\/jats:tex-math>H<\/mml:mi>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>for any sequence of finite-dimensional subspaces$$({{\\mathcal {H}}}_N)_{N=1}^\\infty $$<\/jats:tex-math>(<\/mml:mo>H<\/mml:mi>N<\/mml:mi><\/mml:msub>)<\/mml:mo><\/mml:mrow>N<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow>\u221e<\/mml:mi><\/mml:msubsup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>that is asymptotically dense in$$H^{1\/2}(\\Gamma )$$<\/jats:tex-math>H<\/mml:mi>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Long-standing open questions are whether the essential norm is also$$<1\/2$$<\/jats:tex-math><<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>forD<\/jats:italic>as an operator on$$L^2(\\Gamma )$$<\/jats:tex-math>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>for all Lipschitz$$\\Gamma $$<\/jats:tex-math>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in 2-d; or whether, for all Lipschitz$$\\Gamma $$<\/jats:tex-math>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>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$$\\pm \\frac{1}{2}I+D$$<\/jats:tex-math>\u00b1<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac>I<\/mml:mi>+<\/mml:mo>D<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are compact perturbations of coercive operators\u2014this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces$$({{\\mathcal {H}}}_N)_{N=1}^\\infty $$<\/jats:tex-math>(<\/mml:mo>H<\/mml:mi>N<\/mml:mi><\/mml:msub>)<\/mml:mo><\/mml:mrow>N<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow>\u221e<\/mml:mi><\/mml:msubsup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>that is asymptotically dense in$$L^2(\\Gamma )$$<\/jats:tex-math>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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 ofD<\/jats:italic>is$$\\ge 1\/2$$<\/jats:tex-math>\u2265<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and examples with Lipschitz constant two for which the operators$$\\pm \\frac{1}{2}I +D$$<\/jats:tex-math>\u00b1<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac>I<\/mml:mi>+<\/mml:mo>D<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are not coercive plus compact. We also give, for every$$C>0$$<\/jats:tex-math>C<\/mml:mi>><\/mml:mo>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, examples of Lipschitz polyhedra for which the essential norm is$$\\ge C$$<\/jats:tex-math>\u2265<\/mml:mo>C<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and for which$$\\lambda I+D$$<\/jats:tex-math>\u03bb<\/mml:mi>I<\/mml:mi>+<\/mml:mo>D<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is not a compact perturbation of a coercive operator for any real or complex$$\\lambda $$<\/jats:tex-math>\u03bb<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with$$|\\lambda |\\le C$$<\/jats:tex-math>|<\/mml:mo>\u03bb<\/mml:mi>|<\/mml:mo>\u2264<\/mml:mo>C<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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$$L^2(\\Gamma )$$<\/jats:tex-math>L<\/mml:mi>2<\/mml:mn><\/mml:msup>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>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$$C(\\Gamma )$$<\/jats:tex-math>C<\/mml:mi>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, equivalent to the standard supremum norm, for which the essential norm ofD<\/jats:italic>on$$C(\\Gamma )$$<\/jats:tex-math>C<\/mml:mi>(<\/mml:mo>\u0393<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is$$<1\/2$$<\/jats:tex-math><<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00211-021-01256-x","type":"journal-article","created":{"date-parts":[[2021,12,24]],"date-time":"2021-12-24T16:02:50Z","timestamp":1640361770000},"page":"299-371","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains"],"prefix":"10.1007","volume":"150","author":[{"given":"S. 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