Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations
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1995
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Resumo |
We consider second kind integral equations of the form x(s) - (abbreviated x - K x = y ), in which Ω is some unbounded subset of Rn. Let Xp denote the weighted space of functions x continuous on Ω and satisfying x (s) = O(|s|-p ),s → ∞We show that if the kernel k(s,t) decays like |s — t|-q as |s — t| → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∈ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∈ B(X0) then (I - K)-1 ∈ B(XP) for 0 < p < q, and (I- K)-1∈ B(Xq) if further conditions on k hold. Thus, if k(s, t) = O(|s — t|-q). |s — t| → ∞, and y(s)=O(|s|-p), s → ∞, the asymptotic behaviour of the solution x may be estimated as x (s) = O(|s|-r), |s| → ∞, r := min(p, q). The case when k(s,t) = к(s — t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I — K, for I — K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propaga-tion above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation mod-els, in particular, road traffic noise propagation along an infinite road surface sur-rounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground. |
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text |
Identificador |
http://centaur.reading.ac.uk/32663/1/euclid.jiea.1181075881.pdf Chandler-Wilde, S. N. <http://centaur.reading.ac.uk/view/creators/90000890.html> and Peplow, A. T. (1995) Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations. Journal of Integral Equations and Applications, 7 (3). pp. 303-327. ISSN 1938-2626 doi: 10.1216/jiea/1181075881 <http://dx.doi.org/10.1216/jiea/1181075881> |
Idioma(s) |
en |
Publicador |
Rocky Mountain Mathematics Consortium |
Relação |
http://centaur.reading.ac.uk/32663/ creatorInternal Chandler-Wilde, Simon N. http://dx.doi.org/10.1216/jiea/1181075881 10.1216/jiea/1181075881 |
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Article PeerReviewed |