Mappings of domains connected with the Dirichlet problem for the equation of vibrating string (Q1340381)

This paper is concerned with the solvability of the Dirichlet problem for the vibrating string equation \(u_{xx} - u_{yy} + f(x,y,u) = 0\), \((x,y) \in \Omega\) with \(u |_{\partial \Omega} = 0\), in which \(\Omega\) is a bounded domain with piecewise smooth boundaries. The problem is re-written in a simpler form of a type first noticed by Fritz John, and the result is used to establish existence, uniquenesss and regularity for weak solutions under certain conditions. The main objective of the paper is to derive necessary and sufficient conditions for the functions used when mapping the problem into one involving the simpler form.