Abstract
A class of canonical wedge diffraction problems for Helmholtz equations was formulated by E. Meister in 1986 and subsequently treated by an operator theoretical approach in various publications of his research group including two of the authors. Certain subclasses of those problems, recognized of being unsolved, are subject of the present paper. Some of them are now solved explicitly by refined operator theoretical and analytical methods, others are reduced to systems of equations which contain so-called convolution type operators with symmetry. By a new factorization approach those are proved to be Fredholm in certain (fractional) Sobolev spaces, sometimes with necessary compatibility conditions. Several of the associated operators are therefore explicitly inverted and a number of new problems can be recognized reflecting the challenges of the present state-of-the-art.
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References
H. Bart, V.E. Tsekanovskii, Matricial coupling and equivalence after extension. T. Ando et al. (eds.), Operator Theory and Complex Analysis. Birkhäuser, Basel. Oper. Theory, Adv. Appl. 59 (1992), 143–160.
E.L. Basor, T. Ehrhardt, On a class of Toeplitz + Hankel operators. New York J. Math. 5 (1999), 1–16.
L.P. Castro, R. Duduchava, F.-O. Speck, Localization and minimal normalization of some basic mixed boundary value problems. S. Samko et al. (eds.), Factorization, Singular Operators and Related Problems, Kluwer Academic Publishers, Dordrecht, 2003, 73–100.
L.P. Castro, F.-O. Speck, Regularity properties and generalized inverses of delta-related operators. Z. Anal. Anwend. 17 (1998), 577–598.
[5] L.P. Castro, F.-O. Speck, F.S. Teixeira, A direct approach to convolution type operators with symmetry. To appear.
M. Costabel, E. Stephan, A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985), 367–413.
G.I. Eskin, Boundary value problems for elliptic pseudodifferential equations. Translations of Mathematical Monographs 52. AMS, Providence, R.I., 1981.
I. Gohberg, M.A. Kaashoek, I.M. Spitkovsky, An overview of matrix factorization theory and operator applications. I. Gohberg et al. (eds.), Factorization and Integrable Systems, Proceedings of the Summer School, Faro, Portugal, 2000. Operator Theory: Advances and Applications, 141. Birkhäuser, Basel, 2003, 1–102.
A.B. Lebre, A. Moura Santos, F.-O. Speck, Factorization of a class of matrices generated by Sommerfeld diffraction problems with oblique derivatives. Math. Meth. Appl. Sci. 20 (1997), 1185–1198.
E. Meister, Some solved and unsolved canonical problems of diffraction theory. Differential Equations and their Applications, Equadiff 6, Lect. Notes Math. 1192 (1986), 393–398.
E. Meister, Some solved and unsolved canonical problems of diffraction theory. Differential Equations and Mathematical Physics, Lect. Notes Math. 1285 (1987), 320–336.
E. Meister, Einige gelöste and ungelöste kanonische Probleme der mathematischen Beugungstheorie. Expo. Math. 5 (1987), 193–237.
E. Meister (ed.), Modern mathematical methods in diffraction theory and its applications in engineering. Proceedings of the Sommerfeld’96 workshop. Freudenstadt, Germany. Methoden und Verfahren der Mathematischen Physik. 42. Peter Lang, Frankfurt, Europ. Verlag der Wissenschaften, 1997.
E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two-media scattering problems in a half-space. H. Begehr et al. (eds.), Partial Differential Equations with Real Analysis. Longman Scientific & Technical, Harlow. Pitman Res. Notes Math. Ser. 263 (1992), 122–146.
E. Meister, F. Penzel, F.-O. Speck, 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), 275–294.
E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two canonical wedge problems for the Helmholtz equation. Math. Methods Appl. Sci. 17 (1994), 877–899.
E. Meister, P.A. Santos, F.S. Teixeira, A Sommerfeld-type diffraction problem with second-order boundary conditions. Z. Angew. Math. Mech. 72 (1992), 621–630.
E. Meister, F.-O. Speck, Diffraction problems with impedance conditions. Appl. Anal. 22 (1986), 193–211.
E. Meister, F.-O. Speck, A contribution to the quarter-plane problem in diffraction theory. J. Math. Anal. Appl. 130 (1988), 223–236.
E. Meister, F.-O. Speck, Modern Wiener-Hopf methods in diffraction theory. Ordinary and Partial Differential Equations, Vol. II, Proc. 10th Dundee Conf., Pitman Res. Notes Math. Ser. 216 (1989), 130–171.
A. Moura Santos, F.-O. Speck, Sommerfeld diffraction problems with oblique derivatives. Math. Methods Appl. Sci. 20 (1997), 635–652.
A. Moura Santos, F.-O. Speck, F.S. Teixeira, Minimal normalization of Wiener-Hopf operators in spaces of Bessel potentials. J. Math. Anal. Appl. 225 (1998), 501–531.
F. Penzel, F.S. Teixeira, The Helmholtz equation in a quadrant with Robin’s conditions. Math. Methods Appl. Sci. 22 (1999), 201–216.
A.H. Serbest (ed.), S.R. Cloude (ed.), Direct and inverse electromagnetic scattering. Proceedings of the Workshop in Gebze/Istanbul, Turkey. Pitman Research Notes in Mathematics Series 361. Longman, Harlow, 1996.
A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), 317–374.
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.
F.-O. Speck, Sommerfeld diffraction problems with first and second kind boundary conditions. SIAM J. Math. Anal. 20 (1989), 396–407.
F.-O. Speck, R.A. Hurd, E. Meister, Sommerfeld diffraction problems with third kind boundary conditions. SIAM J. Math. Anal. 20 (1989), 589–607.
H. Triebel, Theory of function spaces II. Monographs in Mathematics 84. Birkhäuser Verlag, Basel, 1992.
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Castro, L.P., Speck, FO., Teixeira, F.S. (2004). On a Class of Wedge Diffraction Problems posted by Erhard Meister. In: Gohberg, I., Wendland, W., Ferreira dos Santos, A., Speck, FO., Teixeira, F.S. (eds) Operator Theoretical Methods and Applications to Mathematical Physics. Operator Theory: Advances and Applications, vol 147. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7926-2_27
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