Arkiv för Matematik

  • Ark. Mat.
  • Volume 42, Number 1 (2004), 119-152.

Fundamental solutions of the acoustic and diffusion equations in nonhomogeneous medium

Victor P. Palamodov

Full-text: Open access

Abstract

A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction.

Note

Partially supported by a stipend of the Mathematical Scientific Research Institute at Berkeley, 2001.

Article information

Source
Ark. Mat., Volume 42, Number 1 (2004), 119-152.

Dates
Received: 20 November 2002
Revised: 7 March 2003
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898841

Digital Object Identifier
doi:10.1007/BF02432913

Mathematical Reviews number (MathSciNet)
MR2056548

Zentralblatt MATH identifier
1088.35018

Rights
2004 © Institut Mittag-Leffler

Citation

Palamodov, Victor P. Fundamental solutions of the acoustic and diffusion equations in nonhomogeneous medium. Ark. Mat. 42 (2004), no. 1, 119--152. doi:10.1007/BF02432913. https://projecteuclid.org/euclid.afm/1485898841


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References

  • Arridge, S. R., Optical tomography in medical imaging, Inverse Problems 15 (1999), R41-R93.
  • Babich, V. M. and Buldyrev, V. S., Short-Wavelength Diffraction Theory. Asymptotic Methods, Springer-Verlag, Berlin, 1991.
  • Bishop, R. L. and Crittenden, R. J., Geometry of Manifolds, Academic Press, New York-London, 1964.
  • Burq, N., Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not. 5 (2002), 221–241.
  • Duistermaat, J. J. and Hörmander, L., Fourier integral operators II, Acta Math. 128 (1972), 183–269.
  • Eidus, D. M., The principle of limiting absorption, Mat. Sb. 57 (99), (1962), 13–44 (Russian). English transl.: Amer. Math. Soc. Transl. 47 (1965), 157–191.
  • Hadamard, J., Le problème de Cauchy et les équations aux dérivées partielles linéares hyperboliques, Hermann, Paris, 1932.
  • Palamodov, V. P., Stability in diffraction tomography and a nonlinear “basic theorem”, J. Anal. Math. 91 (2003), 247–268.
  • Sobolev, S. L., Méthode nouvelle à resoudre le problème de Cauchy pour les équations linéares hyperboliques normales. Mat. Sb. 1 (43) (1936), 39–72.