Electron. J. Diff. Equ., Vol. 2013 (2013), No. 247, pp. 1-21.

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

Guo-Chun Wen

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.

Submitted November 1, 2013. Published November 15, 2013.
Math Subject Classifications: 35J56, 35J25, 35J60, 35B45.
Key Words: Well-posedness; discontinuous boundary value problem; nonlinear elliptic complex equation; A priori estimate; existence of solutions.

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Guo-Chun Wen
LMAM, School of Mathematical Sciences
Peking University
Beijing 100871, China
email: Wengc@math.pku.edu.cn

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