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The operators in these formulations are both continuous and coercive on general Lipschitz domains in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, in the space <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2(\\Gamma )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>\u0393<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299\u2013371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace\u2019s equation (involving the double-layer potential and its adjoint) <jats:italic>cannot<\/jats:italic> be written as the sum of a coercive operator and a compact operator in the space <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2(\\Gamma )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>\u0393<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in <jats:inline-formula><jats:alternatives><jats:tex-math>$${L^2(\\Gamma )}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>\u0393<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> do <jats:italic>not<\/jats:italic> converge when applied to the standard second-kind formulations, but <jats:italic>do<\/jats:italic> converge for the new formulations.\n<\/jats:p>","DOI":"10.1007\/s00211-024-01424-9","type":"journal-article","created":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T19:01:51Z","timestamp":1718737311000},"page":"1325-1384","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains"],"prefix":"10.1007","volume":"156","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0578-1283","authenticated-orcid":false,"given":"S. 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