A problem for the Helmholtz equation outside cuts on the plane with the Dirichlet condition and the oblique derivative condition on opposite sides of the cuts (Q656490)

Let \(\Gamma\) denotes a finite set of simple open curves on the plane \(\mathbb R^2\). The authors consider the Helmholtz equation \(\Delta u+ k^2u= 0\), \(k=\text{const}\neq 0\), \(\arg k\in [0,\pi/2)\cup (\pi/2,\pi)\) in \(\mathbb R^2\setminus\Gamma\). The Dirichlet condition is posed on one side of each curve from \(\Gamma\), and an oblique derivative condition with pure imaginary coefficient \(\beta\) of the tangential derivative is posed on the other side. In addition, suitable conditions are given at infinity. The uniqueness of the solution is proved. For \(|\beta|<1\), the solvability to the problem is obtained and an integral representation in the form of potentials of the solution is found.