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On Some Boundary Value Problems for the Helmholtz Equation in a Cone of 240º

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Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 221))

Abstract

In this paper we present the explicit solution in closed analytic form of Dirichlet and Neumann problems for the Helmholtz equation in the non-convex and non-rectangular cone Ω0,α with α = 4π/3. Actually, these problems are the only known cases of exterior (i.e., α > π) wedge diffraction problems explicitly solvable in closed analytic form with the present method. To accomplish that, we reduce the BVPs in Ω0,α each to a pair of BVPs with symmetry in the same cone and each BVP with symmetry to a pair of semi-homogeneous BVPs in the convex half-cones. Since α/2 is an (odd) integer part of 2π, we obtain the explicit solution of the semi-homogeneous BVPs for half-cones by so-called Sommerfeld potentials (resulting from special Sommerfeld problems which are explicitly solvable).

Mathematics Subject Classification (2000). Primary 35J25; Secondary 30E25, 35J05, 45E10, 47B35, 47G30.

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References

  1. L.P. Castro and D. Kapanadze, Exterior wedge diffraction problems with Dirichlet, Neumann and impedance boundary conditions. Acta Appl. Math. 110 (2010), 289-311.

    Article  MathSciNet  MATH  Google Scholar 

  2. L.P. Castro and F.-O. Speck, Regularity properties and generalized inverses of delta-related operators. Z. Anal. Anwend. 17 (1998), 577-598.

    MathSciNet  MATH  Google Scholar 

  3. L.P. Castro, F.-O. Speck and F.S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister. Oper. Theory Adv. Appl. 147 (2004), 211-238.

    MathSciNet  Google Scholar 

  4. L.P. Castro, F.-O. Speck and F.S. Teixeira, Mixed boundary value problems for the Helmholtz equation in a quadrant. Integr. Equ. Oper. Theory 56 (2006), 1-44.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985), 367-413.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Ehrhardt, A.P. Nolasco and F.-O. Speck, Boundary integral methods for wedge diffraction problems: the angle 2π/n, Dirichlet and Neumann conditions, Operators and Matrices, 5 (2011), 1-40.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Ehrhardt, A.P. Nolasco and F.-O. Speck, A Riemannn surface approach for diffraction from rational wedges, to appear.

    Google Scholar 

  8. G.I. Èskin, Boundary Value Problems for Elliptic Pseudodifferential Equations Translations of Mathematical Monographs 52, AMS, 1981.

    Google Scholar 

  9. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 24, Pitman, 1985.

    Google Scholar 

  10. A.E. Heins, The Sommerfeld half-plane problem revisited, II, the factoring of a matrix of analytic functions. Math. Meth. Appl. Sci. 5 (1983), 14-21.

    Article  MathSciNet  MATH  Google Scholar 

  11. G.C. Hsiao and W.L. Wendland, Boundary Integral Equations. Applied Mathematical Sciences Series 164, Springer-Verlag, 2008.

    Google Scholar 

  12. A.I. Komech, N.J. Mauser and A.E. Merzon, On Sommerfeld representation and uniqueness in scattering by wedges. Math. Meth. Appl. Sci. 28 (2005), 147-183.

    Article  MathSciNet  MATH  Google Scholar 

  13. G.D. Malujinetz, Excitation, reflection and emission of the surface waves on a wedge with given impedances of the sides. Dokl. Acad. Nauk SSSR 121 (1958), 436-439.

    Google Scholar 

  14. E. Meister, Ein Überblick über analytische Methoden zur Lösung singulärer Integralgleichungen. Proceedings of the GAMM Conference in Graz in 1976, Z. Angew. Math. Mech. 57 (1977), T23-T35.

    MathSciNet  MATH  Google Scholar 

  15. E. Meister, Some multiple-part Wiener-Hopf problems in Mathematical Physics. Mathematical Models and Methods in Mechanics, Banach Center Publications 15,PWN–Polish Scientific Publishers, Warsaw (1985), 359-407.

    Google Scholar 

  16. E. Meister, Some solved and unsolved canonical problems of diffraction theory. Lecture Notes in Math. 1285, (1987) 320-336.

    Article  MathSciNet  Google Scholar 

  17. E. Meister, F. Penzel, F.-O. Speck and F.S. Teixeira, Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant. Proc. R. Soc. Edinb., Sect. A 123 (1993), 193-237.

    Article  MathSciNet  Google Scholar 

  18. E. Meister, P.A. Santos and F.S. Teixeira, A Sommerfeld-type diffraction problem with second-order boundary conditions. Z. Angew. Math. Mech. 72 (1992), 621-630.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Meister and F.-O. Speck, Modern Wiener-Hopf methods in diffraction theory. Pitman Res. Notes Math. Ser. 216 (1989), 130-171.

    MathSciNet  Google Scholar 

  20. A. Moura Santos, F.-O. Speck and F.S. Teixeira, Minimal normalization ofWiener-Hopf operators in spaces of Bessel potentials. J. Math. Anal. Appl. 225 (1998), 501-531.

    Article  MathSciNet  MATH  Google Scholar 

  21. A.V. Osipov and A.N. Norris, The Malyuzhinets theory for scattering from wedge boundaries: a review. Wave Motion 29 (1999), 313-340.

    Article  MathSciNet  MATH  Google Scholar 

  22. A.D. Rawlins, The explicit Wiener-Hopf factorization of a special matrix.Z.Angew. Math. Mech. 61 (1981), 527-528.

    MathSciNet  MATH  Google Scholar 

  23. A.D. Rawlins, The solution of a mixed boundary value problem in the theory of diffraction. J.Eng.Math. 18 (1984), 37-62.

    Article  MathSciNet  MATH  Google Scholar 

  24. A.D. Rawlins, Plane-wave diffraction by a rational angle. Proc. R. Soc. Lond. A 411 (1987), 265-283.

    Article  MATH  Google Scholar 

  25. A.F. dos Santos, A.B. Lebre and F.S. Teixeira, The diffraction problem for a half-plane with different face impedances revisited. J. Math. Anal. Appl. 140 (1989), 485-509.

    Article  MathSciNet  MATH  Google Scholar 

  26. F.-O. Speck, Mixed boundary value problems of the type of Sommerfeld’s half-plane problem. Proc. R. Soc. Edinb., Sect. A 104 (1986), 261-277.

    Article  MathSciNet  MATH  Google Scholar 

  27. P.Ya. Ufimtsev, Theory of Edge Diffraction in Electromagnetics. Tech Science Press, 2003.

    Google Scholar 

  28. P. Zhevandrov and A. Merzon, On the Neumann problem for the Helmholtz equation in a plane angle. Math. Meth. Appl. Sci. 23 (2000), 1401-1446.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    A. P. Nolasco

  2. Department of Mathematics, I.S.T. Technical University of Lisbon, 1049-001, Lisbon, Portugal

    F.-O. Speck

Authors
  1. A. P. Nolasco
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  2. F.-O. Speck
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Corresponding author

Correspondence to A. P. Nolasco .

Editor information

Editors and Affiliations

  1. , Abt. Mathematik V, Universität Ulm, Ulm, 89069, Germany

    Wolfgang Arendt

  2. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  3. Institut für Mathematik, AG Modellierung, Numerik und, TU Berlin, Straße des 17 Juni 136, Berlin, 10623, Berlin, Germany

    Jussi Behrndt

  4. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Karl-Heinz Förster

  5. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  6. , Institut für Mathematik, TU Ilmenau, Weimarer Str. 25, Ilmenau, 98693, Thüringen, Germany

    Carsten Trunk

Additional information

Dedicated to Professor Erhard Meister on the occasion of his 80th anniversary

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Nolasco, A.P., Speck, FO. (2012). On Some Boundary Value Problems for the Helmholtz Equation in a Cone of 240º. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_30

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