The oblique derivative problem for second order nonlinear equations of mixed type with two degenerate lines

Abstract

In [1–10], the authors posed and discussed the Tricomi problem of second order equations of mixed (elliptic-hyperbolic) type, which possesses important applications to gas dynamics, but they only consider some special equations of mixed type. In [5, 7], the authors proposed and discussed the Tricomi problem for some second order equations of mixed type with nonsmooth degenerate line under stronger conditions. The present paper deals with the oblique derivative problem for second order nonlinear equations of mixed type with two parabolic degenerate lines in a plane domain. The boundary value problem includes the Tricomi problem of Chaplygin equation as a special case. Firstly the existence of solutions of corresponding boundary value problems for degenerate elliptic and hyperbolic equations of second order are discussed, and then the solvability of the oblique derivative problem for the equations of mixed type with two degenerate lines is proved. The used method in this paper is different to those in above papers, because the equations are nonlinear, and the new notations are introduced, such that the second order equations of mixed type with degenerate lines are reduced to the first order mixed complex equations with singular coefficients, then the advantage of complex analytic method can be applied, and we can see that the method can be used to handle more general problems. Finally we mention that the authors in [21–24] discussed oblique derivative problems for the Helmholtz and Laplace equations in the general interior and exterior domains.