Inverse problem for equations of mixed type with Lavrent’ev-Bitsadze operator

Abstract

For the equation of mixed elliptic-hyperbolic type $$u_{xx} + (\operatorname{sgn} y)u_{yy} - b^2 u = f(x)$$ in a rectangular domainD = {(x, y) | 0 < x < 1, −α < y < β}, where α, β, and b are given positive numbers, we study the problem with boundary conditions $$\begin{gathered} u(0,y) = u(1,y) = 0, - \alpha \leqslant y \leqslant \beta , \hfill \\ u(x,\beta ) = \phi (x),u(x,\alpha ) = \psi (x),u_y (x, - \alpha ) = g(x),0 \leqslant x \leqslant 1. \hfill \\ \end{gathered} $$ . We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution.