The extended Bitsadze-Lavrent’ev-Tricomi boundary value problem

F. G. Tricomi (5], 6]) originated the theory of boundary of value problems for mixed type equations by establishing the first mixed type equation known asthe Tricomi equation \(y \cdot u_{xx} + u_{yy} = 0\) which is hyperbolic fory<0, elliptic fory>0, and parabolic fory=0 and then observed that this equation could be applied in Aerodynamics and in general in Fluid Dynamics (transonic flows). See: M. Cribario 1], G. Fichera 2], and our doctoral dissertation 4]. Then M. A. Lavrent’ev and A. V. Bitsadze 3] established together a new mixed type boundary value problem for the equation \(\operatorname{sgn} (y) \cdot u_{xx} + u_{yy} = 0\) where sgn (y)=1 fory>0, =?1 fory<0, fory=0, which involved thediscontinuous coefficient K=sgn (y) ofu _xx while in the case of Tricomi equation the corresponding coefficientT=y wascontinuous. In this paper we establish another mixed type boundary value problem forthe extended Bitsadze-Lavrent’ev-Tricomi equation \(L u = \operatorname{sgn} (y) \cdot u_{xx} + \operatorname{sgn} (x) \cdot u_{yy} + r (x,y) \cdot u = f (x,y)\) where both coefficientsK=sgn (y),M=sgn (x) ofu _xx ,u _yy , respectively are discontinous,r=r (x, y) is once continuously differentiable,f=f (x, y) continuous, and then we prove a uniqueness theorem for quasi-regular solutions.