半平面衍射问题索末菲解唯一性的时间相关方法
Time-dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half-plane
摘要：
我们认为索末菲问题的衍射由一个不透明的半平面与一个真正的波数解释它的极限情况下，随着时间趋于无穷大，相应的时间相关的衍射问题。我们证明了Sommerfeld公式是该问题解的极限振幅，该解属于某个泛函类且在该泛函类中是唯一的.为了证明该问题解的唯一性，我们将其在$t$中进行Fourier-Laplace变换后，化为一个复波数的定常衍射问题.这允许我们使用Sobolev空间中唯一性的证明$H^1$。因此，我们避免从一开始就在边缘上施加辐射和正则性条件，而是以自然的方式获得它。
英文摘要：
We consider the Sommerfeld problem of diffraction by an opaque half-plane with a real wavenumber interpreting it as the limiting case, as time tends to infinity, of the corresponding time-dependent diffraction problem. We prove that the Sommerfeld formula for the solution is the limiting amplitude of the solution of this time-dependent problem which belongs to a certain functional class and is unique in it. For the proof of uniqueness of solution to the time-dependent problem we reduce it, after the Fourier-Laplace transform in $t$, to a stationary diffraction problem with a complex wavenumber. This permits us to use the proof of uniqueness in the Sobolev space $H^1$. Thus we avoid imposing the radiation and regularity conditions on the edge from the beginning and instead obtain it in a natural way.
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