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A version of Runge's theorem for the Helmholtz equation with applications to scattering theory*

Published online by Cambridge University Press:  20 January 2009

R. L. Ochs Jr
R. L. Ochs Jr
Affiliation:
Department of MathematicsThe University of ToledoToledoOhio 43606, USA
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Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinates

where C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equation

in D can be approximated in the space by an entire Herglotz wave function

with kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 1 , February 1989 , pp. 107 - 119
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).Google Scholar
2.Bowman, F., Introduction to Bessel Functions (Dover, New York, 1958).Google Scholar
3.Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983).Google Scholar
4.Colton, D. and Monk, P., A novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region, SIAM J. Appl. Math. 45 (1985), 1039–1053.CrossRefGoogle Scholar
5.Colton, D. and Monk, P., A novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region II, SIAM J. Appl. Math. 46 (1986), 506–523.CrossRefGoogle Scholar
6.Colton, D. and Monk, P., The numerical solution of the three dimensional inverse scattering problem for time harmonic acoustic waves, SIAM J. Sci. Statist. Comput. 8 (1987), 278–291.CrossRefGoogle Scholar
7.Colton, D. and Monk, P., The inverse scattering problem for time harmonic acoustic waves in a penetrable medium, Quart. J. Mech. Appl. Math. 40 (1987), 189–212.CrossRefGoogle Scholar
8.Colton, D. and Monk, P., The inverse scattering problem for time harmonic acoustic waves in an inhomogeneous medium, Quart. J. Appl. Math., 41 (1988), 97–125.CrossRefGoogle Scholar
9.Colton, D. and Wimp, J., Generalized Herglotz domains, Math. Meth. Appl. Sci. 8 (1986), 451–457.CrossRefGoogle Scholar
10.Hartman, P. and Wilcox, C., On solutions of the Helmholtz equation in exterior domains, Math. Z. 75 (1961), 228–255.CrossRefGoogle Scholar
11.Müller, C., Über die ganzen Lösungen der Wellengleichung, Math. Ann. 124 (1952), 235–264.CrossRefGoogle Scholar
12.Müller, C., Angewandte Mathematik, Gustav Herglotz: Gesammelte Schriften (Schwerdtfeger, Hans, ed., Vandenhoeck and Ruprecht, Göttingen, xxix–xxxi, 1979).Google Scholar
13.Ochs, R. L. Jr., The limited aperture problem of inverse acoustic scattering (Ph.D. dissertation, University of Delaware, Newark, Delaware, 1986).Google Scholar
14.Ochs, R. L. Jr., The limited aperture problem of inverse acoustic scattering: Dirichlet boundary conditions, SIAM J. Appl. Math. 47 (1987), 1320–1341.CrossRefGoogle Scholar
15.Pironneau, O., Optimal Shape Design for Elliptic Systems (Springer-Verlag, New York, 1984).CrossRefGoogle Scholar
16.Vekua, I. N., New Methods for Solving Elliptic Equations (North-Holland, Amsterdam 1967).Google Scholar