The Riemann-Hilbert problem for mixed complex equations of first order with degenerate rank 0. (English) Zbl 1315.35136
Summary: This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed (elliptic-hyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of Hölder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.
MSC:
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35M30 | Mixed-type systems of PDEs |
| 35M32 | Boundary value problems for mixed-type systems of PDEs |
| 35J56 | Boundary value problems for first-order elliptic systems |
| 35L40 | First-order hyperbolic systems |
Keywords:
Riemann-Hilbert problem; quasilinear mixed complex equations; degenerate rank 0; unique solvability of the problem; Hölder continuity of singular double integerReferences:
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