Nonlinear Riemann–Hilbert problems for quasilinear -equations on the unit disc

Abstract

The existence of solutions of nonlinear Riemann–Hilbert problems for quasilinear -equations on the unit disc is considered. Let be a smooth function with bounded first derivatives and let be a family of Jordan curves in . Let be a smooth solution of the equation on such that belongs to the interior of the Jordan curve for every . Then there exists a smooth solution u of the equation on such that for every . Moreover, there is a sequence of solutions of this Riemann–Hilbert boundary value problem which uniformly on compact subsets of converges to .