Diffraction problems associated with either electromagnetic or acoustic waves in the half space \(\mathbb{R}_ +^ 2\) are considered. \(\mathbb{R}_ +^ 2\) is divided into two quadrants by the interface (\(x_ 1=0\), \(x_ 2>0\)) and in each quadrant the wave number is different. Dirichlet, Neumann or mixed conditions are imposed on the boundary \((\pm x_ 1>0\), \(x_ 2=0)\). Such problems are shown to be wellposed in the energy space \(H_ 1\).
After commenting on the approaches adopted by other workers the authors present a new operator theoretic method for solving such problems with the overall aim of obtaining explicit analytic representations for the scattered fields. This objective is achieved for Dirichlet and Neumann problems. However for the mixed problem it was only possible to reduce the problem to a vectorial Riemann-Hilbert problem in \([L_ 2^ + (\mathbb{R})]^ 2\) whose explicit resolution relies on a, presently unavailable, factorisation of a piecewise continuous symbol. Despite this, asymptotic behaviour near the origin, for plane wave excitation, was achieved.
For the entire collection see [Zbl 0771.00049].