The New Well-Posedness of Discontinuous Riemann-Hilbert Problem for Elliptic Complex Equations of First Order With Two Degenerate Lines

Abstract

This article deals with some elliptic complex equations of first order, i.e. the generalized Beltrimi equation with two degenerate lines in the discussed multiply connected domain. We first propose the new well-posed-ness of discontinuous Riemman-Hilbert problem, give estimates of solutions for the modified boundary value problem. Afterwards by using the method of parameter extension, the existence of continuous solutions for the generalized Beltrimi equation is verified. In the article, the proof of Hölder continuity of a singular double integer is very difficult and interesting. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. AMS Subject Classification: 35J60, 35J70, 35J56Key Words: elliptic complex equations, two degenerate lines, the new posedness of discontinuous Riemann-Hilbert problem, unique solvability of continuous solutions, Hölder continuity of singular integer