Well-posedness of discontinuous boundary-value problems for nonlinear elliptic complex equations in multiply connected domains

Abstract

In the first part of this article, we study a discontinuous Riemann-Hilbert problem for nonlinear uniformly elliptic complex equations of first order in multiply connected domains. First we show its well-posedness. Then we give the representation of solutions for a modified Riemann-Hilbert problem for the complex equations. Then we obtain a priori estimates of the solutions and verify the solvability of the modified problem by using the Leray-Schauder theorem. Then the solvability of the original discontinuous Riemann-Hilbert boundary-value problem is obtained. In the second part, we study a discontinuous Poincare boundary-value problem for nonlinear elliptic equations of second order in multiply connected domains. First we formulate the boundary-value problem and show its new well-posedness. Next we obtain the representation of solutions and obtain a priori estimates for the solutions of a modified Poincare problem. Then with estimates and the method of parameter extension, we obtain the solvability of the discontinuous Poincare problem.

Keywords

• Well-posedness
• discontinuous boundary value problem
• nonlinear elliptic complex equation
• A priori estimate
• existence of solutions