an:00179404
Zbl 0779.35063
G??rard, Patrick; Lebeau, Gilles
Diffraction of a wave at a corner
FR
J. Am. Math. Soc. 6, No. 2, 341-424 (1993).
00013168
1993
j
35L05 35S15 35A20 78A45
boundary-value problem; singularities; diffraction of a cylindrical wave; curved wedge
The authors consider the boundary-value problem given by
\[
(\partial^ 2_ t -\Delta)u=0, \qquad u|_{\partial\Omega}=0, \qquad u|_{t<0}= u_ i| _{t<0},
\]
where \(u_ i\equiv u_ i(t,x,y)\in H_{\text{loc}}'(\mathbb{R}\times \mathbb{R}^ 2)\) is a certain given function satisfying \((\partial^ 2_ t-\Delta)u_ i=0\) and \(\Omega\subset \mathbb{R}\times \mathbb{R}^ 2\) is an open domain defined by \(\Omega= \mathbb{R}\times(\mathbb{R}^ 2 \setminus F)\) with \(F=\{(x,y)\in\mathbb{R}^ 2\), \(x\geq 0\), \(b(x)\leq y\leq a(x)\}\), and study the singularities of the solution for \(t>0\). Here \(a(x)\in C^ \infty\) and \(b(x)\in C^ \infty\) are two given functions, analytic near the origin, such that \(a(x)>0>b(x)\) when \(x>0\) while \(a(0)=b(0)=0\) and \(a'(0)>0>b'(0)\). This problem corresponds to the diffraction of a cylindrical wave (namely: \(u_ 1\)) by the curved wedge \(W=\{(x,y,z)\mid z\in(-\infty,\infty)\), \((x,y)\in F\}\).
The paper is rather long and includes three theorems, eleven propositions and eighteen lemmas. It is worthwile to notice that the title of the paper is misleading in physical point of view because the diffraction does not occur here at a cone tip but rather at an edge.
M.Idemen (??stanbul)