On boundary value problems for the hyperbolic case (Q1336483)

The situation for the solvability and the uniqueness of the Dirichlet problem for the wave equation depends on the form of the domain and the boundary values. This dependence is not of a natural kind dealing with smoothness properties of functions describing the boundary and the data. For example, if the domain is a rectangle \(0 \leq x \leq X\), \(0 \leq y \leq Y\) then the character of the problem depends on arithmetical properties of the ratio \(\lambda = X/Y\). In Section 2 of the paper the author determines the approximate solution of the Dirichlet problem for the vibrating string equation for any \(\lambda\) and shows that the accuracy of the approximation is bounded from below for rational \(\lambda\), while there is no positive lower bound for irrational \(\lambda\). In Section 3 the author considers the boundary value problem with the prescribed skew derivative on the boundary \(\Gamma_ R\) (the boundary of the rectangle \(R\) given by \(0 \leq x \leq X\), \(0 \leq y \leq Y)\), for the vibrating string equation in a rectangle \(R\). The last section of the paper contains the Dirichlet problem for a mixed, elliptic-hyperbolic type equation (the Lavrentiev-Bitsadze equation) given by \[ {\partial^ 2 u \over \partial x^ 2} + \text{sgn} y {\partial^ 2 u \over \partial y^ 2} = 0. \]