Weighted Sobolev Spaces and Exterior Problems for the Helmholtz Equation
Abstract
Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the RellichSommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form (partial/partial r  ik) u in L_2(Ω), r = x in Ω subset R^n, it is shown here that radiating solutions satisfy a condition of the form (1+r)^1/2 (ln (e + r))^{1/2 δ} u in L_2(Ω), 0 < δ < 1/2, and, moreover, such solutions satisfy the classical Sommerfeld condition u = O(r^{1/2(n1)}), r > ∞. Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincare inequality and the Sobolev embedding theorems.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 April 1987
 DOI:
 10.1098/rspa.1987.0044
 Bibcode:
 1987RSPSA.410..373N