Electron. J. Diff. Equ.,
Vol. 2013 (2013), No. 247, pp. 121.
Wellposedness of discontinuous boundaryvalue problems for
nonlinear elliptic complex equations in multiply connected domains
GuoChun Wen
Abstract:
In the first part of this article, we study a discontinuous RiemannHilbert
problem for nonlinear uniformly elliptic complex equations of first order
in multiply connected domains. First we show its wellposedness.
Then we give the representation of solutions for a modified RiemannHilbert
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 LeraySchauder theorem. Then the solvability
of the original discontinuous RiemannHilbert boundaryvalue
problem is obtained. In the second part, we study a discontinuous
Poincare boundaryvalue problem for nonlinear elliptic equations
of second order in multiply connected domains.
First we formulate the boundaryvalue problem and show its new wellposedness.
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: Wellposedness; discontinuous boundary value problem;
nonlinear elliptic complex equation; A priori estimate;
existence of solutions.
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GuoChun Wen
LMAM, School of Mathematical Sciences
Peking University
Beijing 100871, China
email: Wengc@math.pku.edu.cn

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