Weighted Sobolev spaces and exterior problems for the Helmholtz equation
Weighted Sobolev spaces and exterior problems for the Helmholtz equation
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亥姆霍兹方程的加权索博列夫空间和外部问题
DOI：
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
G. Roach
中科院分区：
文献类型：
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作者：
P. Neittaanmäki;G. Roach
加权索博列夫空间用于解决亥姆霍兹方程外部问题解的存在性和唯一性问题。此外，研究表明，这种方法可以解决问题中的非齐次项，只需在无穷远处渐近消失。与 Rellich-Sommerfeld 辐射条件相反，在希尔伯特空间设置中，要求亥姆霍兹方程的所有辐射解应满足 (∂/∂r−ik)u∈L2(Ω),r=|x|εΩ⊂Rn 形式的条件，此处显示辐射解满足以下形式的条件(1+r)−12(ln(e+r))−12δu∈L2(Ω),0<δ<12，而且，这样的解满足经典索末菲条件 u=O(r−12(n−1)),r→∞。此外，该方法避免了通常与庞加莱不等式和索博列夫嵌入定理的应用相关的许多困难。
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−ik)u∈L2(Ω),r=|x|∈Ω⊂Rn, it is shown here that radiating solutions satisfy a condition of the form (1+r)−12(ln(e+r))−12δu∈L2(Ω),0<δ<12, and, moreover, such solutions satisfy the classical Sommerfeld condition u=O(r−12(n−1)),r→∞. Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.