{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T20:31:58Z","timestamp":1759177918523},"reference-count":65,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T00:00:00Z","timestamp":1683072000000},"content-version":"vor","delay-in-days":32,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2023,4]]},"abstract":"Abstract<\/jats:title>We say that $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $$x\\in \\Gamma $$<\/jats:tex-math>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \u0393<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is either locally $$C^1$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or locally coincides (in some coordinate system centred at x<\/jats:italic>) with a Lipschitz graph $$\\Gamma _x$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that $$\\Gamma _x=\\alpha _x\\Gamma _x$$<\/jats:tex-math>\n \n \n \u0393<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n \u03b1<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n \n \u0393<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for some $$\\alpha _x\\in (0,1)$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n x<\/mml:mi>\n <\/mml:msub>\n \u2208<\/mml:mo>\n \n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we study, for such $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the essential spectrum of $$D_\\Gamma $$<\/jats:tex-math>\n \n D<\/mml:mi>\n \u0393<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the double-layer (or Neumann\u2013Poincar\u00e9) operator of potential theory, on $$L^2(\\Gamma )$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u0393<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show, via localisation and Floquet\u2013Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators $$K_t$$<\/jats:tex-math>\n \n K<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for $$t\\in [-\\pi ,\\pi ]$$<\/jats:tex-math>\n \n t<\/mml:mi>\n \u2208<\/mml:mo>\n [<\/mml:mo>\n -<\/mml:mo>\n \u03c0<\/mml:mi>\n ,<\/mml:mo>\n \u03c0<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; moreover, each $$K_t$$<\/jats:tex-math>\n \n K<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is compact if $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$C^1$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> except at finitely many points. For the 2D case where, additionally, $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of $$D_\\Gamma $$<\/jats:tex-math>\n \n D<\/mml:mi>\n \u0393<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nystr\u00f6m-method approximations to the operators $$K_t$$<\/jats:tex-math>\n \n K<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfies the well-known spectral radius conjecture, that the essential spectral radius of $$D_\\Gamma $$<\/jats:tex-math>\n \n D<\/mml:mi>\n \u0393<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on $$L^2(\\Gamma )$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u0393<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$<1\/2$$<\/jats:tex-math>\n \n <<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all Lipschitz $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We illustrate this theory with examples; for each we show that the essential spectral radius is<\/jats:italic>$$<1\/2$$<\/jats:tex-math>\n \n <<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal $$C^{1,\\beta }$$<\/jats:tex-math>\n \n C<\/mml:mi>\n \n 1<\/mml:mn>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.\n<\/jats:p>","DOI":"10.1007\/s00211-023-01353-z","type":"journal-article","created":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T17:01:57Z","timestamp":1683133317000},"page":"635-699","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains"],"prefix":"10.1007","volume":"153","author":[{"given":"Simon N.","family":"Chandler-Wilde","sequence":"first","affiliation":[]},{"given":"Raffael","family":"Hagger","sequence":"additional","affiliation":[]},{"given":"Karl-Mikael","family":"Perfekt","sequence":"additional","affiliation":[]},{"given":"Jani A.","family":"Virtanen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,3]]},"reference":[{"key":"1353_CR1","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s00205-015-0928-0","volume":"220","author":"H Ammari","year":"2016","unstructured":"Ammari, H., Deng, Y., Millien, P.: Surface plasmon resonance of nanoparticles and applications in imaging. 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