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 Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ 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 ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.

Well-Posed PDE and Integral Equation Formulations for Scattering by Fractal Screens

“…The following well-posedness is classical, and can be established by combining the equivalence of formulations in Remark 3.4 with results in [56,Corollary 4.5] for SN-w, in [45] for SD-w. In each case uniqueness follows from Green's first theorem and a result of Rellich (cf.…”

“…These are long-standing scattering problems, their mathematical study dating back at least to [49, p. 139], and it is well-known (e.g., [45,56], and see §3.1 for more detail) that, for arbitrary bounded Γ ⊂ Γ ∞ , these problems are well-posed (and the solutions depend only on the closure Γ) if the boundary conditions are understood in the standard weak senses that u ∈ W 1,loc 0 (D) in the sound-soft case, that u ∈ W 1,loc (D) and D (v∆u + ∇v · ∇u) dx = 0, for all v ∈ W 1,comp (D), (1.3) in the sound-hard case. We spell out these weak formulations more fully in Definitions 3.1 and 3.2 below using standard Sobolev space notations defined in §2.…”