E. Meister et F. Penzel, 2 CANONICAL WEDGE PROBLEMS FOR THE HELMHOLTZ-EQUATION, Mathematical methods in the applied sciences, 17(11), 1994, pp. 877-899
Boundary-transmission problems for two-dimensional Helmholtz equations
in a quadrant and its complement, respectively, are considered in a S
obolev space setting. The first problem of a quadrant with Dirichlet c
ondition on one face and transmission condition on the other is solved
in closed form for the case where all the quadrants are occupied by t
he same medium. Unique solvability can also be shown in the case of tw
o different media up to exceptional cases of wave numbers, while the F
redholm property holds in general. In the second problem, transmission
conditions are prescribed on both faces. Similar results are obtained
in the one-medium case, but the two-media case turns out to be more c
omplicated and the equivalent system of boundary pseudodifferential eq
uations cannot be completely analysed by this reasoning.