## 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

Zbl 0354.35013
Full Text:

### References:

 [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 [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). [6] V. M. Babich, ”On the shortwave asymptotics for Green’s function of the Helmholtz equation,” Mat. Sb.,65(107), No. 4, 576–630 (1964). [7] V. D. Andronov, ”Some estimates of Green’s function for the Helmholtz equation,” Zh. Vychisl. Mat. Mat. Fiz.,5, No. 6, 1006–1023 (1965). · Zbl 0162.42401 [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 [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 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.