The diffraction problem for a half plane with different face impedances revisited
In the complement of a ``screen'' \(\{x\in {\mathbb{R}}^ 3|\) \(x_ 2=0\wedge x_ 1\geq 0\}\) the authors consider a special boundary value problem for Maxwell's equation which may be reduced to an interface problem for Helmholtz's equation \((\Delta +k^ 2)\phi =0\) in \({\mathbb{R}}^ 2\) with certain transmission conditions at \(x_ 1=0\). Only the case of a dissipative medium is treated (i.e. Im k\(>0)\) therefore the condition at infinity may be formulated as \(\lim \phi (x_ 1,x_ 2)=0\) as \(| x_ 2| \to \infty\). If the desired solution \(\phi\) is viewed upon as an element \(\phi (\cdot,x_ 2)\in {\mathcal S}'({\mathbb{R}})\) an equivalent Wiener-Hopf formulation may be derived. The authors show that the theory of singular integral operators may be applied successfully to this system of Wiener-Hopf equations once the right Sobolev spaces \(H^{\alpha}\) are chosen for \(\phi (\cdot,x_ 2)\).
boundary value problem for Maxwell's equation
interface problem for Helmholtz's equation
scientific article; zbMATH DE number 4134886