A class of nonlinear riemann-hilbert problems for the second order elliptic system

Abstract

Nonlinear Riemann-Hilbert problems for first and second order elliptic system have been studied by many authors. They usually apply Schauder's fixed point theorem to prove the existence of a Hölder continuous solution in the Sobolev space . In [1, 2] L.V. Wolfersdorf discusses a class of nonlinear Riemann-Hilbert problems for holomorphic functions with monotone nonlinearity and proves the existence of a solution in W1 2. In [3] R.P Gilbert and Li Mingzhong further discuss the existence of solution from W1 2 for generalized analytic functions and first-order nonlinear elliptic system where the proof of existence of a solution in W1 2 has been established. In this paper, we continue our study of nonlinear Riemann-Hilbert problems for second order elliptic system. Using an explicit form for the solution, a reduction is made to a nonlinear boundary-value problem for two holomorphic functions. Using an approximation we are led to consider a solvable perturbed problem. A limiting procedure with suitable a priori estimates, permit the proof of the existence of a solution in