×

On some asymptotic representations of solutions to elliptic equations and their applications. (English) Zbl 1467.35114

The paper presents asymptotic representations of solutions to elliptic boundary value problems depending on a complex parameter and suggests some applications to inverse problems.

MSC:

35J08 Green’s functions for elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R30 Inverse problems for PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Vainberg, BR., Asymptotic expansion as \(####\) of the solutions of exterior boundary value problems for hyperbolic equations and quasiclassical approximations, Partial Differ Equ V Encycl Math Sci, 34, 53-89 (1999)
[2] Vainberg, BR., On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as \(####\) of solutions of non-stationary problems, Russ Math Surv, 30, 1-58 (1975) · Zbl 0318.35006
[3] Babich, VM., The multidimensional WKB method or ray method. Its analogs and generalizations, Itogi Nauki Tekh Ser Sovrem Probl Mat Fundam Napravleniya, 34, 93-134 (1988) · Zbl 0657.35119
[4] Babich, VM; Buldyrev, VS., Asymptotic methods in diffraction problems of short waves (1972), Moskau: Nauka, Moskau
[5] Babich, VM., The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium, USSR Comput Math Math Phys, 5, 247-251 (1965) · Zbl 0167.24301
[6] Babich, VM., On short wave asymptotic form of Green’s function for the Helmholtz equation, Mat Sb N Ser, 65, 107, 576-630 (1964)
[7] Buslaev, VS., Potential theory and geometrical optics, J Sov Math, 2, 204-209 (1974) · Zbl 0284.35017
[8] Olimpiev, IV., Estimates of the field in the shadow region in diffraction of a cylindrical wave at a bounded convex cylinder, Sov Phys Dokl, 9, 441-443 (1964) · Zbl 0137.07604
[9] Vainberg, BR., On the analytical properties of the resolvent for a certain class of operator-pencils, Math USSR-Sb, 6, 241-273 (1968) · Zbl 0186.20804
[10] Vainberg, BR., On a point source in an inhomogeneous medium, Math USSR, 23, 23-148 (1974)
[11] Gataullin, TM., Asymptotic behavior of the fundamental solution of an elliptic equation with respect to a complex parameter, Math Notes, 21, 210-217 (1977) · Zbl 0399.35012
[12] Babich, VM; Kisilev, AP., Elastic waves: high frequency theory (2018), CRS Press: Boca Raton, CRS Press
[13] Marchuk, GI.Mathematical models in environmental problems. Amsterdam: Elsevier Science Publishers; 1986. (Studies in Mathematics and its Applications; Vol. 16). · Zbl 0597.90001
[14] Badia, AEl; Ha-Duong, T., An inverse source problem in potential analysis, Inverse Probl, 16, 651-663 (2000) · Zbl 0963.35194
[15] Ling, L.; Takeuchi, T., Point sources identification problems for heat equations, Commun Comput Phys, 5, 897-913 (2009) · Zbl 1364.35436
[16] Pyatkov, SG; Safonov, EI., Point sources recovering problems for the one-dimensional heat equation, J Adv Res Dyn Control Syst, 11, 496-510 (2019)
[17] Triebel, H., Interpolation theory, function spaces, differential operators (1978), Berlin: VEB Deutscher Verlag der Wissenschaften, Berlin · Zbl 0387.46033
[18] Vladimirov, VS; Zharinov, VV., Equations of mathematical physics (2004), Moscow: Fizmatlit, Moscow
[19] Sveshnikov, AG; Bogolyubov, AN; Kravtsov, VV., Lectures on mathematical physics (1993), Moscow: Iz-vo MGU, Moscow
[20] Watson, GN., A treatize of the theory of Bessel functions (1944), Cambridge: Cambridge University Press, Cambridge
[21] Amann, H.Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function spaces, differential operators and nonlinear analysis. Teubner-Texte Math. Bd. 133. Stuttgart: Teubner; 1993. p. 9-126.
[22] Amann, H., Nonautonomous parabolic equations involving measures, J Math Sci, 30, 4780-4802 (2005) · Zbl 1159.35386
[23] Amann, H.Maximum principles and principal eigenvalues. In: Ferrera J, Lopez-Gomez J, Ruiz del Portal FR, editors. Ten Mathematical Essays on Approximation in Analysis and Topology. Amsterdam: Elsevier; 2005. p. 1-60. · Zbl 1090.35090
[24] Denk, R.; Hieber, M.; Prüss, J., R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem Amer Math Soc, 166 (2003) · Zbl 1274.35002
[25] Kato, T., Perturbation theory for linear operators (1995), Berlin: Springer-Verlag, Berlin · Zbl 0836.47009
[26] Hieber, M.; Schrohe, L., \(####\) spectral independence of elliptic operators via commutator estimates, Positivity, 3, 259-272 (1999) · Zbl 0930.35058
[27] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order (2001), Berlin: Springer-Verlag, Berlin · Zbl 1042.35002
[28] Courant, R., Methods of mathematical physics, Vol. 2 (1962), New York (NY): John Wiley & Sons, New York (NY)
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.