Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Authors:
L. P. Castro and D. Kapanadze

Journal:
Proc. Amer. Math. Soc. **136** (2008), 2113-2123

MSC (2000):
Primary 35J25; Secondary 35J05, 45E10, 47A20, 47G30, 47H50

DOI:
https://doi.org/10.1090/S0002-9939-08-09288-5

Published electronically:
February 14, 2008

MathSciNet review:
2383517

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

**1.**Bair Budaev,*Diffraction by wedges*, Pitman Research Notes in Mathematics Series, vol. 322, Longman Scientific & Technical, Harlow, 1995. MR**1413497****2.**L.P. Castro and D. Kapanadze,*Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case*,

Math. Nachr., to appear, 12 pp.**3.**L. P. Castro and D. Kapanadze,*On wave diffraction by a half-plane with different face impedances*, Math. Methods Appl. Sci.**30**(2007), no. 5, 513–527. MR**2298679**, https://doi.org/10.1002/mma.794**4.**Luis P. Castro and David Natroshvili,*The potential method for the reactance wave diffraction problem in a scale of spaces*, Georgian Math. J.**13**(2006), no. 2, 251–260. MR**2252591****5.**L. P. Castro, F.-O. Speck, and F. S. Teixeira,*Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip*, J. Integral Equations Appl.**15**(2003), no. 4, 359–383. MR**2058809**, https://doi.org/10.1216/jiea/1181074982**6.**Luis P. Castro, Frank-Olme Speck, and Francisco S. Teixeira,*On a class of wedge diffraction problems posted by Erhard Meister*, Operator theoretical methods and applications to mathematical physics, Oper. Theory Adv. Appl., vol. 147, Birkhäuser, Basel, 2004, pp. 213–240. MR**2053691****7.**L. P. Castro, F.-O. Speck, and F. S. Teixeira,*Mixed boundary value problems for the Helmholtz equation in a quadrant*, Integral Equations Operator Theory**56**(2006), no. 1, 1–44. MR**2256995**, https://doi.org/10.1007/s00020-005-1410-4**8.**David Colton and Rainer Kress,*Inverse acoustic and electromagnetic scattering theory*, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR**1635980****9.**Martin Costabel and Ernst Stephan,*A direct boundary integral equation method for transmission problems*, J. Math. Anal. Appl.**106**(1985), no. 2, 367–413. MR**782799**, https://doi.org/10.1016/0022-247X(85)90118-0**10.**Jean-Pierre Croisille and Gilles Lebeau,*Diffraction by an immersed elastic wedge*, Lecture Notes in Mathematics, vol. 1723, Springer-Verlag, Berlin, 1999. MR**1740860****11.**A. I. Komech, N. J. Mauser, and A. E. Merzon,*On Sommerfeld representation and uniqueness in scattering by wedges*, Math. Methods Appl. Sci.**28**(2005), no. 2, 147–183. MR**2110262**, https://doi.org/10.1002/mma.553**12.**A. K. Gautesen,*Diffraction of plane waves by a wedge with impedance boundary conditions*, Wave Motion**41**(2005), no. 3, 239–246. MR**2120170**, https://doi.org/10.1016/j.wavemoti.2004.05.009**13.**G.D. Malyuzhinets,*Excitation, reflection and emission of surface waves from a wedge with given face impedances*(English; Russian original), Sov. Phys., Dokl.**3**(1959), 752-755; translation from Dokl. Akad. Nauk SSSR**121**(1959), 436-439.**14.**William McLean,*Strongly elliptic systems and boundary integral equations*, Cambridge University Press, Cambridge, 2000. MR**1742312****15.**Erhard Meister,*Some solved and unsolved canonical problems of diffraction theory*, Differential equations and mathematical physics (Birmingham, Ala., 1986) Lecture Notes in Math., vol. 1285, Springer, Berlin, 1987, pp. 320–336. MR**921283**, https://doi.org/10.1007/BFb0080611**16.**E. Meister, F.-O. Speck, and F. S. Teixeira,*Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions*, J. Integral Equations Appl.**4**(1992), no. 2, 229–255. MR**1172891**, https://doi.org/10.1216/jiea/1181075683**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. Roy. Soc. Edinburgh Sect. A**123**(1993), no. 2, 275–294. MR**1215413**, https://doi.org/10.1017/S0308210500025671**18.**E. Meister, F. Penzel, F.-O. Speck, and F. S. Teixeira,*Two canonical wedge problems for the Helmholtz equation*, Math. Methods Appl. Sci.**17**(1994), no. 11, 877–899. MR**1289599**, https://doi.org/10.1002/mma.1670171104**19.**E. Meister and F.-O. Speck,*Modern Wiener-Hopf methods in diffraction theory*, Ordinary and partial differential equations, Vol. II (Dundee, 1988) Pitman Res. Notes Math. Ser., vol. 216, Longman Sci. Tech., Harlow, 1989, pp. 130–171. MR**1031728****20.**F. Penzel and F. S. Teixeira,*The Helmholtz equation in a quadrant with Robin’s conditions*, Math. Methods Appl. Sci.**22**(1999), no. 3, 201–216. MR**1672263**, https://doi.org/10.1002/(SICI)1099-1476(199902)22:3<201::AID-MMA32>3.0.CO;2-T**21.**A. D. Rawlins,*A note on point source diffraction by a wedge*, J. Appl. Math.**1**(2004), 85–89. MR**2077882**, https://doi.org/10.1155/S1110757X04306066**22.**S.L. Sobolev,*Partial Differential Equations of Mathematical Physics*

(English translation). New-York: Dover Publications (1989).**23.**A. Sommerfeld,*Mathematische Theorie der Diffraction*, Math. Ann.**47**(1896), no. 2-3, 317–374 (German). MR**1510907**, https://doi.org/10.1007/BF01447273

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35J25,
35J05,
45E10,
47A20,
47G30,
47H50

Retrieve articles in all journals with MSC (2000): 35J25, 35J05, 45E10, 47A20, 47G30, 47H50

Additional Information

**L. P. Castro**

Affiliation:
Research Unit “Mathematics and Applications”, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Email:
castro@ua.pt

**D. Kapanadze**

Affiliation:
Research Unit “Mathematics and Applications”, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Email:
david.kapanadze@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-08-09288-5

Received by editor(s):
March 22, 2007

Published electronically:
February 14, 2008

Additional Notes:
This work was supported in part by Unidade de Investigação Matemática e Aplicações of Universidade de Aveiro, and the Portuguese Science Foundation (FCT–Fundação para a Ciência e a Tecnologia) through grant number SFRH/BPD/20524/2004.

Communicated by:
Hart F. Smith

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.