Existence theorem for the solution of Dirichlet and Neumann problems for the Helmholtz equation in the quasiperiodic case
Consider the Dirichlet and the Neumann quasiperiodic problems for the Helmholtz equation \[ (1)\quad (\Delta +k^ 2)u=0,\quad k>0,\quad (x,y)\in \Omega \subset R^ 2 \] in two different cases of \(\Omega\). The so called quasiperiodicity means that \[ u(x+2\pi,y)=u(x,y)\exp (2\pi i\alpha)\quad \forall (x,y)\in \Omega \] where \(\alpha\) is a fixed real number. The author gives existence theorems for solutions of these problems.
scientific article; zbMATH DE number 4118785