Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip
Let \(\Omega_+\) indicate the first quadrant in \(\mathbb R^2\) and \(\Omega_-\) its complement, \(\Gamma_1:= \{(0,x_2) : x_2 > 0\}\), \(\Gamma_2:= \{(x_1,0) : 0 < x_1 < a\}\), with \(a \in \mathbb R^+\), \(k \in \mathbb C\) with \(\text{Im}(k) > 0\). The authors study the following boundary value problem: \[ \begin{cases} (\Delta + k^2) u = 0 & \text{ in } \Omega_\pm ,\\ u(0^+,x_2) = u(0^-,x_2) = f_+(x_2), & (0, x_2) \in \Gamma_1,\\ {\partial u \over \partial x_2}(x_1,0^+) = {\partial u \over \partial x_2}(x_1,0^-) = - g_a(x_1), & (x_1,0) \in \Gamma_2. \end{cases}\tag{1} \] Here \(f_+ \in H^{s-1/2}(\mathbb R^+)\), \(g_a \in H^{s-3/2}(]0, a[)\) for some \(s \in ]1/2, 3/2[\). A solution of (1) is (by definition) an element \(u \in L^2(\mathbb R^2)\) such that \(u_{\Omega^\pm} \in H^s(\Omega^\pm)\). The boundary conditions are intended in a generalized sense. Problem (1) models a rectangular wedge diffraction problem. The authors are able to prove a well-posedness result and to write down the solution explicitly in terms of inverse matrixes involving Wiener-Hopf and Wiener-Hopf-Hankel operators.
scientific article; zbMATH DE number 2114498
scientific article