Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations


Autoria(s): Chandler-Wilde, Simon N.; Peplow, Andrew T.
Data(s)

1995

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.

Formato

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

Tipo

Article

PeerReviewed