Asymptotics of solutions to the wave equation in a domain with a small hole

Korikov, D.

doi:10.1090/spmj/1360

Asymptotics of solutions to the wave equation in a domain with a small hole
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by D. V. Korikov
Translated by: the author

St. Petersburg Math. J. 26 (2015), 813-838

DOI: https://doi.org/10.1090/spmj/1360

Published electronically: July 27, 2015

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Abstract:

In a bounded domain with a small hole (at all times $t\in \mathbb {R}$), the wave equation is considered with homogeneous Dirichlet condition on the boundary. The asymptotics of the solution as the diameter of the hole tends to 0 is deduced. To describe the behavior of long waves, the method of compound asymptotic expansions is used. The contribution of short waves (the wavelength is smaller than the diameter of the hole) to the energy of the solution is negligible due to the smoothness of the right-hand side of the wave equation with respect to time.

References

Vladimir Maz′ya, Serguei Nazarov, and Boris Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000. Translated from the German by Georg Heinig and Christian Posthoff. MR 1779977
M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk 19 (1964), no. 3 (117), 53–161 (Russian). MR 0192188
B. A. Plamenevskiĭ, On the Dirichlet problem for the wave equation in a cylinder with edges, Algebra i Analiz 10 (1998), no. 2, 197–228 (Russian); English transl., St. Petersburg Math. J. 10 (1999), no. 2, 373–397. MR 1629407
A. Yu. Kokotov and B. A. Plamenevskiĭ, On the asymptotic behavior of solutions of the Neumann problem for hyperbolic systems in domains with conical points, Algebra i Analiz 16 (2004), no. 3, 56–98 (Russian); English transl., St. Petersburg Math. J. 16 (2005), no. 3, 477–506. MR 2083566, DOI 10.1090/S1061-0022-05-00862-9
S. I. Matyukevich and B. A. Plamenevskiĭ, On dynamic problems in the theory of elasticity in domains with edges, Algebra i Analiz 18 (2006), no. 3, 158–233 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 3, 459–510. MR 2255852, DOI 10.1090/S1061-0022-07-00957-0
A. M. Il′in, A boundary value problem with a small parameter, Uspehi Mat. Nauk 32 (1977), no. 3 (195), 161–162 (Russian). MR 0460871
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
S. A. Nazarov and B. A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries, Nauka, Moscow, 1991. (Russian).
V. G. Maz′ja and B. A. Plamenevskiĭ, Estimates in $L_{p}$ and in Hölder classes, and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr. 81 (1978), 25–82 (Russian). MR 492821, DOI 10.1002/mana.19780810103