On the construction of eigenfunctions of the adjoint Tricomi problem for the Lavrent'ev-Bitsadze operator (Q1177469)

The author studies the spectral problem \[ u_{xx}+\text{sgn} yu_{yy}+\lambda u=0 \qquad ((x,y)\in D_ +\cup D_ -); \] \(u(x,y)\in C(\overline{D})\cap C^ 1(D)\cap C^ 2(D_ +\cup D_ -)\); \(u(x,y)=0\) \(((x,y)\in\overline {CB\cup\sigma})\), where \(D\subset\mathbb{R}^ 2\) is a domain bounded by the curve \(\sigma\) lying in the half-plane \(y>0\) and the segments \(AC\) and \(CB\) (\(A=(0,0)\), \(B=(1,0)\), \(C=(1/2,-1/2)\)); \(D_ +=D\cap\{y>0\}\), \(D_ -=D\cap\{y<0\}\). The author reduces this problem to a certain elliptic spectral problem in \(D_ +\). If \(D_ +\) is a sector that is bounded by the radii \(BA\) and \(BK\) and the arc \(AK\), then the author gives the explicit solution of this problem.