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Ursell’s approach to obtaining an a priori estimate for the solution of the Neumann problem for the Helmholtz equation. (English) Zbl 0396.35005
MSC:
35B45 A priori estimates in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35A08 Fundamental solutions to PDEs
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[1] V. M. Babich, ”On the rigorous justification of the shortwave approximation in the three-dimensional case,” in: Mathematical Questions in the Theory of Wave Propagation, Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,34, 23–52 (1973).
[2] V. M. Babich, ”The method of D. Ludwig and the method of the boundary layer in the problem of diffraction by a smooth body,” in: Boundary-Value Problems of Mathematical Physics and Related Questions in Function Theory, Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, No. 6, 17–33 (1972).
[3] D. Ludwig, ”Uniform asymptotic expansion of the field scattered by a convex object at high frequencies,” Comm. Pure Appl. Math.,20, No. 1, 103–180 (1967). · Zbl 0154.12802 · doi:10.1002/cpa.3160200103
[4] V. S. Buslaev, ”Potential theory and geometrical optics,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,22, 175–180 (1971). · Zbl 0284.35017
[5] F. Ursell, ”On the shortwave asymptotic theory of the wave equation (\(\Delta\)+K2)=0,” Proc. Cambr. Phil. Soc.,53, No. 1, 115–133 (1957). · doi:10.1017/S0305004100032060
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[8] V. D. Andronov, ”On the shortwave asymptotics of the solution to the Neumann problem in the case of the Helmholtz equation,” in: Numerical Methods for Solving Problems of Mathematical Physics [in Russian], Nauka (1966), pp. 144–153.
[9] V. S. Buslaev, ”On the shortwave asymptotics in problems of diffraction by convex bodies,” Dokl. Akad. Nauk SSSR,145, No. 4, 753–756 (1962).
[10] F. Ursell, ”On the rigorous foundation of shortwave asymptotics,” Proc. Cambr. Phil. Soc.,62, No. 2, 227–244 (1966). · Zbl 0142.45503 · doi:10.1017/S0305004100039797
[11] H. Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Springer, Berlin (1953). · Zbl 0050.07402
[12] F. G. Leppington, ”Creeping waves in the shadow of an elliptic cylinder,” J. Inst. Math. Applic.,3, No. 4, 388–402 (1967). · Zbl 0153.56503 · doi:10.1093/imamat/3.4.388
[13] N. S. Grigor’eva, ”Uniform asymptotic expansions of functions related to a paraboloid of revolution,” in: Mathematical Questions in the Theory of Wave Propagation, Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,25, No. 4, 52–78 (1972).
[14] V. M. Babich, ”Finding the saddle point in the case of the problem on the ellipse,” in: Mathematical Questions in the Theory of Wave Propagation, Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,17, 20–24 (1970).
[15] M. F. Fedoryuk, ”The method of stationary phase for multidimensional integrals,” Zh. Vychisl. Mat. Mat. Fiz.,2, No. 1, 145–150 (1962). · Zbl 0122.12401
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