Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators

  • Jeffrey Galkowski
  • Euan A. SpenceEmail author
Open Access


We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from \({L^2({\partial {\Omega }})}\rightarrow H^1({\partial {\Omega }})\) (where \({\partial {\Omega }}\) is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in \({L^2({\partial {\Omega }})}\), of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit \({L^2({\partial {\Omega }})}\rightarrow H^1({\partial {\Omega }})\) bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper (Galkowski, Müller, and Spence in Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem, 2017. arXiv:1608.01035).


Helmholtz equation Layer-potential operators High frequency Semiclassical Boundary integral equation 

Mathematics Subject Classification

31B10 31B25 35J05 35J25 65R20 



  1. 1.
    Anand, A., Boubendir, Y., Ecevit, F., Reitich, F.: Analysis of multiple scattering iterations for high-frequency scattering problems. II: the three-dimensional scalar case. Numerische Mathematik 114(3), 373–427 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Asheim, A., Huybrechs, D.: Extraction of uniformly accurate phase functions across smooth shadow boundaries in high frequency scattering problems. SIAM J. Appl. Math. 74(2), 454–476 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge Monographs on Applied and Computational Mathematics (1997)Google Scholar
  4. 4.
    Baskin, D., Spence, E.A., Wunsch, J.: Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations. SIAM J. Math. Anal. 48(1), 229–267 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Burq, N., Gérard, P., Tzvetkov, N.: Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds. Duke Math. J. 138(3), 445–486 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Lindner, M.: Condition number estimates for combined potential boundary integral operators in acoustic scattering. J. Integral Equ. Appl. 21(2), 229–279 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numerica 21(1), 89–305 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chandler-Wilde, S.N., Hewett, D.P.: Wavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screens. Integral Equ. Oper. Theory 82(3), 423–449 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chandler-Wilde, S.N., Hewett, D.P., Langdon, S., Twigger, A.: A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numerische Mathematik 129(4), 647–689 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chandler-Wilde, S.N., Hewett, D.P., Moiola, A.: Sobolev spaces on non-Lipschitz subsets of \(\mathbb{R}^n\) with application to boundary integral equations on fractal screens. Integral Equ. Oper. Theory 87(2), 179–224 (2017)zbMATHGoogle Scholar
  11. 11.
    Chandler-Wilde, S.N., Langdon, S.: A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45(2), 610–640 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chandler-Wilde, S.N., Monk, P.: Wave-number-explicit bounds in time-harmonic scattering. SIAM J. Math. Anal. 39(5), 1428–1455 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chandler-Wilde, S.N., Spence, E.A., Gibbs, A., Smyshlyaev, V.P.: High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis. (2017) arXiv preprint. arXiv:1708.08415
  14. 14.
    Christianson, H., Hassell, A., Toth, J.A.: Exterior Mass estimates and \(L^2\)-restriction bounds for Neumann data along hypersurfaces. Int. Math. Res. Notices 6, 1638–1665 (2015)zbMATHGoogle Scholar
  15. 15.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer (1998)Google Scholar
  16. 16.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Domínguez, V., Graham, I.G., Smyshlyaev, V.P.: A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numerische Mathematik 106(3), 471–510 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances. Book in progress (2018).
  19. 19.
    Ecevit, F.: Frequency independent solvability of surface scattering problems. Turkish J. Math. 42(2), 407–417 (2018)MathSciNetGoogle Scholar
  20. 20.
    Ecevit, F., Eruslu, H.H.: A Galerkin BEM for high-frequency scattering problems based on frequency-dependent changes of variables. IMA J. Numer. Anal. to appear (2018)Google Scholar
  21. 21.
    Ecevit, F., Özen, H.Ç.: Frequency-adapted galerkin boundary element methods for convex scattering problems. Numerische Mathematik 135(1), 27–71 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ecevit, F., Reitich, F.: Analysis of multiple scattering iterations for high-frequency scattering problems. Part I: the two-dimensional case. Numerische Mathematik 114, 271–354 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Fabes, E.B., Jodeit, M., Riviere, N.M.: Potential techniques for boundary value problems on \(C^1\) domains. Acta Mathematica 141(1), 165–186 (1978)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Galkowski, J.: Distribution of resonances in scattering by thin barriers. arXiv preprint. arXiv:1404.3709 (to appear in Memoirs of the AMS) (2014)
  25. 25.
    Galkowski, J., Müller, E.H., Spence, E.A.: Wavenumber-explicit analysis for the Helmholtz \(h\)-BEM: error estimates and iteration counts for the Dirichlet problem (2017). arXiv:1608.01035
  26. 26.
    Galkowski, J., Smith, H.F.: Restriction bounds for the free resolvent and resonances in lossy scattering. Int. Math. Res. Notices 16, 7473–7509 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gander, M.J., Graham, I.G., Spence, E.A.: Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik 131(3), 567–614 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ganesh, M., Hawkins, S.: A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions. J. Comput. Phys. 230, 104–125 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Graham, I.G., Löhndorf, M., Melenk, J.M., Spence, E.A.: When is the error in the \(h\)-BEM for solving the Helmholtz equation bounded independently of \(k\)? BIT Numer. Math. 55(1), 171–214 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Greenleaf, A., Seeger, A.: Fourier integral operators with fold singularities. J. Reine Angew. Math. 455, 35–56 (1994)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Han, X., Tacy, M.: Sharp norm estimates of layer potentials and operators at high frequency. J. Funct. Anal. 269(9), 2890–2926 (2015). with an appendix by J. GalkowskiMathSciNetzbMATHGoogle Scholar
  32. 32.
    Hassell, A., Tacy, M.: Semiclassical \(L^p\) estimates of quasimodes on curved hypersurfaces. J. Geom. Anal. 22(1), 74–89 (2012)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hewett, D.P.: Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering. Eur. J. Appl. Math. 26(05), 773–793 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hewett, D.P., Langdon, S., Chandler-Wilde, S.N.: A frequency-independent boundary element method for scattering by two-dimensional screens and apertures. IMA J. Numer. Anal. 35(4), 1698–1728 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hewett, D.P., Langdon, S., Melenk, J.M.: A high frequency hp boundary element method for scattering by convex polygons. SIAM J. Numer. Anal. 51(1), 629–653 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer (1985)Google Scholar
  37. 37.
    Ikawa, M.: Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier 38(2), 113–146 (1988)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kirsch, A.: Surface gradients and continuity properties for some integral operators in classical scattering theory. Math. Methods Appl. Sci. 11(6), 789–804 (1989)MathSciNetzbMATHGoogle Scholar
  39. 39.
    McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  40. 40.
    Melrose, R.B., Sjöstrand, J.: Singularities of boundary value problems. II. Commun. Pure Appl. Math. 35(2), 129–168 (1982)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Melrose, R.B., Taylor, M.E.: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55(3), 242–315 (1985)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Meyer, Y., Coifman, R.: Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  43. 43.
    Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163(2), 181–251 (1999)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Moiola, A., Spence, E.A.: Is the Helmholtz equation really sign-indefinite? SIAM Rev. 56(2), 274–312 (2014)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Morawetz, C.S.: Decay for solutions of the exterior problem for the wave equation. Commun. Pure Appl. Math. 28(2), 229–264 (1975)MathSciNetzbMATHGoogle Scholar
  46. 46.
    NIST. Digital Library of Mathematical Functions (2018).
  47. 47.
    Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer, Berlin (2011)zbMATHGoogle Scholar
  48. 48.
    Spence, E.A.: Wavenumber-explicit bounds in time-harmonic acoustic scattering. SIAM J. Math. Anal. 46(4), 2987–3024 (2014)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Spence, E.A., Chandler-Wilde, S.N., Graham, I.G., Smyshlyaev, V.P.: A new frequency-uniform coercive boundary integral equation for acoustic scattering. Commun. Pure Appl. Math. 64(10), 1384–1415 (2011)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Spence, E.A., Kamotski, I.V., Smyshlyaev, V.P.: Coercivity of combined boundary integral equations in high-frequency scattering. Commun. Pure Appl. Math. 68(9), 1587–1639 (2015)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, New York (2008)zbMATHGoogle Scholar
  52. 52.
    Tacy, M.: Semiclassical \(L^p\) estimates of quasimodes on submanifolds. Commun. Partial Differ. Equ. 35(8), 1538–1562 (2010)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Tacy, M.: Semiclassical \(L^{2}\) estimates for restrictions of the quantisation of normal velocity to interior hypersurfaces. arXiv preprint (2014). arxiv:1403.6575
  54. 54.
    Tataru, D.: On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1), 185–206 (1998)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Vainberg, B.R.: On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as \(t\rightarrow \infty \) of solutions of non-stationary problems. Russian Mathematical Surveys 30(2), 1–58 (1975)zbMATHGoogle Scholar
  56. 56.
    Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. Journal of Functional Analysis 59(3), 572–611 (1984)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Zworski, M.: Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2012)Google Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations