Abstract
We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge.
References
Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York (1972)
Alù A., Silveirinha M.G., Salandrino A., Engheta N.: Epsilon-near zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern. Phys. Rev. B 75, 155410 (2007)
Ammari H., Millien P., Ruiz M., Zhang H.: Mathematical analysis of plasmonic nanoparticles: the scalar case. Arch. Ration. Mech. Anal. 224(2), 597–658 (2017)
Ammari H., Ruiz M., Yu S., Zhang H.: Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations. J. Differ. Equ. 261(6), 3615–3669 (2016)
Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343. Springer, Heidelberg (2011)
Bonnetier, E., Zhang, H.: Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences. Rev. Mat. Iberoam (to appear). arXiv:1702.08127
Chandler-Wilde S.N., Hewett D.P., Moiola A.: Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counter examples. Mathematika 61(2), 414–443 (2015)
Costabel M.: Some historical remarks on the positivity of boundary integral operators, Boundary element analysis Lecture Notes in Applied and Computational Mechanics, Vol. 29. Springer, Berlin (2007) 1–27, 2007
Costabel M., Stephan E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106(2), 367–413 (1985)
Dahlberg B.E.J., Kenig C.E.: Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. (2) 125(3), 437–465 (1987)
Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Ding Z.: A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124(2), 591–600 (1996)
Dobrzynski L., Maradudin A.A.: Electrostatic edge modes in a dielectric wedge. Phys. Rev. B 6, 3810–3815 (1972)
Duflo M., Moore C.C.: On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21(2), 209–243 (1976)
Einav A., Loss M.: Sharp trace inequalities for fractional Laplacians. Proc. Am. Math. Soc. 140(12), 4209–4216 (2012)
Elschner J.: Asymptotics of solutions to pseudodifferential equations of Mellin type. Math. Nachr. 130(1), 267–305 (1987)
Escauriaza L., Fabes E.B., Verchota G.: On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries. Proc. Am. Math. Soc. 115(4), 1069–1076 (1992)
Escauriaza L., Mitrea M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216(1), 141–171 (2004)
Eymard, P., Terp, M.: La transformation de Fourier et son inverse sur le groupe des ax + b d’un corps local, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II, Lecture Notes in Mathematics , Vol. 739, 207–248
Fabes E.B., Jodeit M. Jr., Rivière N.M.: Potential techniques for boundary value problems on C 1-domains. Acta Math. 141(3-4), 165–186 (1978)
Fabes E.B., Jodeit M. Jr., Lewis J.E.: Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 26(1), 95–114 (1977)
Fabes E., Mendez O., Mitrea M.: Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159(2), 323–368 (1998)
Führ H.: Hausdorff–Young inequalities for group extensions. Can. Math. Bull. 49(4), 549–559 (2006)
Gelfand I., Neumark M.: Unitary representations of the group of linear transformations of the straight line. C. R. (Doklady) Acad. Sci URSS (N.S.) 55, 567–570 (1947)
Grachev, N.V., Maz’ya, V.G.: The Fredholm radius of integral operators of potential theory, Nonlinear equations and variational inequalities. Linear operators and spectral theory (Russian), Probl. Mat. Anal., Vol. 11, 109–133, 251, 1990 (translated in J. Soviet Math. 64(6), 1297–1313, 1993)
Hassi S., Sebestyén Z., de Snoo H.S.V.: On the nonnegativity of operator products. Acta Math. Hungar. 109(1–2), 1–14 (2005)
Helsing J., Perfekt K.-M.: On the polarizability and capacitance of the cube. Appl. Comput. Harmon. Anal. 34(3), 445–468 (2013)
Helsing, J., Perfekt, K.-M.: The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points. J. Math. Pures Appl. 118(9), 235–287 (2018)
Hofmann S., Mitrea M., Taylor M.: Singular integrals and elliptic boundary problems on regular Semmes–Kenig–Toro domains. Int. Math. Res. Not. IMRN 14, 2567–2865 (2010)
Kang H., Lim M., Yu S.: Spectral resolution of the Neumann–Poincaré operator on intersecting disks and analysis of plasmon resonance. Arch. Ration. Mech. Anal. 226(1), 83–115 (2017)
Kenig, C.E.: Recent progress on boundary value problems on Lipschitz domains, Pseu dodifferential operators and applications (Notre Dame, Ind., 1984). Proceedings of the Symposium in Pure Mathematics , Vol. 43. American Mathematical Society, Providence,175–205, 1985
Khalil I.: Sur l’analyse harmonique du groupe affine de la droite. Stud. Math. 51, 139–167 (1974)
Khavinson D., Putinar M., Shapiro H.S.: Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185(1), 143–184 (2007)
Klein A., Russo B.: Sharp inequalities for Weyl operators and Heisenberg groups. Math. Ann. 235(2), 175–194 (1978)
Kleppner A., Lipsman R.L.: The Plancherel formula for group extensions. I. Ann. Sci. École Norm. Sup. (4) 5, 459–516 (1972)
Kleppner A., Lipsman R.L.: The Plancherel formula for group extensions. II. Ann. Sci. École Norm. Sup. (4) 6, 103–132 (1973)
Kozlov V.A., Maz’ya V.G., Rossmann J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, Vol. 52. American Mathematical Society, Providence (1997)
Krein M.G.: Compact linear operators on functional spaces with two norms. Integr. Equ. Oper. Theory 30(2), 140–162 (1998) Translated from the Ukranian, dedicated to the memory of Mark Grigorievich Krein (1907–1989)
Lewis J.E.: Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains. Trans. Am. Math. Soc. 320(1), 53–76 (1990)
Lewis J.E.: A symbolic calculus for layer potentials on C 1 curves and C 1 curvilinear polygons. Proc. Am. Math. Soc. 112(2), 419–427 (1991)
Lewis J.E., Parenti C.: Pseudo differential operators of Mellin type. Commun. Partial Differ. Equ. 8(5), 477–544 (1983)
Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, Vol. I. Springer, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181
Maxwell Garnett J.C.: VII. Colours in metal glasses, in metallic films, and in metallic solutions. II. Philos. Trans. R. Soc. A 205(387-401), 237–288 (1906)
McCarthy John E.: Geometric interpolation between Hilbert spaces. Ark.Mat. 30(2), 321–330 (1992)
Medková D.: The Laplace Equation: Boundary Value Problems on Bounded and Un bounded Lipschitz Domains. Springer, Berlin (2018)
Milton G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)
Mitrea I.: On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons. J. Fourier Anal. Appl. 8(5), 443–487 (2002)
Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque, 344, viii+241 (2012)
Nicorovici N.A., McPhedran R.C., Milton G.W.: Transport properties of a three phase composite material: the square array of coated cylinders. Proc. R. Soc. Lond. A 442(1916), 599–620 (1993)
Nikoshkinen K.I., Lindell I.V.: Image solution for Poisson’s equation in wedge geometry. IEEE. Trans. Antennas Propag. 43(2), 179–187 (1995)
Perfekt K.-M., Putinar M.: Spectral bounds for the Neumann–Poincaré operator on planar domains with corners. J. Anal. Math. 124(1), 39–57 (2014)
Perfekt K.-M., Putinar M.: The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Arch. Ration. Mech. Anal. 223(2), 1019–1033 (2017)
Qiao Y.: Double layer potentials on three-dimensional wedges and pseudodifferential operators on Lie groupoids. J. Appl. Math. Anal. Appl. 462(1), 428–447 (2018)
Qiao Y., Nistor V.: Single and double layer potentials on domains with conical points I: straight cones. Integr. Equ. Oper. Theory 72(3), 419–448 (2012)
Scharstein R.W.: Green’s function for the harmonic potential of the three-dimensional wedge transmission problem. IEEE. Trans. Antennas Propag. 52(2), 452–460 (2004)
Schmüdgen K.: Unbounded Self-Adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, Vol. 265. Springer, Dordrecht (2012)
Shelepov V.Yu.: On the index and spectrum of integral operators of potential type along Radon curves. Mat. Sb. 181(6), 751–778 (1990)
Valagiannopoulos C.A., Sihvola A.: Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration. Radio Sci. 48(3), 316–325 (2013)
Verchota G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)
Wallén H., Kettunen H., Sihvola A.: Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2, 113–121 (2008)
Yu S., Ammari H.: Plasmonic interaction between nanospheres. SIAM Rev. 60(2), 356–385 (2018)
Acknowledgements
The author thanks Johan Helsing, Anders Karlsson, and Tobias Kuna for helpful discussions. The author is also grateful to the American Institute of Mathematics, San Jose, USA and the Erwin Schrödinger Institute, Vienna, Austria, in each of which some of this work was prepared.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Le Bris
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Perfekt, KM. The Transmission Problem on a Three-Dimensional Wedge. Arch Rational Mech Anal 231, 1745–1780 (2019). https://doi.org/10.1007/s00205-018-1308-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1308-3