Application of complex analysis to second order equations of mixed type

This paper deals with an application of complex analysis to second order equations of mixed type. We mainly discuss the discontinuous Poincaré boundary value problem for a second order linear equation of mixed (elliptic-hyperbolic) type, i.e. the generalized Lavrent'ev-Bitsadze equation with weak conditions, using the methods of complex analysis. We first give a representation of solutions for the above boundary value problem, and then give solvability conditions via the Fredholm theorem for integral equations. In [1], [2], the Dirichlet problem (Tricomi problem) for the mixed equation of second order ${\displaystyle u_{\times }+sgny{u}_{yy}=0}$ was investigated. In [3], the Tricomi problem for the generalized Lavrent'ev-Bitsadze equation ${\displaystyle u_{\times }+sgny{u}_{yy}+A{u}_x+B{u}_y+Cu=0}$, i.e. ${\displaystyle u_{\xi \eta }+a{u}_{\xi }+b{u}_{\eta }+cu=0}$ with the conditions: a ≥ 0, ${\displaystyle a_{\xi }+ab-c\ge 0}$, c ≥ 0 was discussed in the hyperbolic domain. In the present paper, we remove the above assumption of [3] and obtain a solvability result for the discontinuous Poincaré problem, which includes the corresponding results in [1]-[3] as special cases.