NONLINEAR RIEMANN-HILBERT PROBLEMS WITHOUT TRANSVERSALITY

Nonlinear Riemann-Hilbert problems (RHP) generalize two fundamental cl assical problems for complex analytic functions, namely: 1. the confor mal mapping problem, and 2. the linear Riemann-Hilbert problem. This p aper presents new results on global existence for the nonlinear (RHP) in doubly connected domains with nonclosed restriction curves for the boundary data. More precisely, our nonlinear (RHP) is required to beco me ''at infinity'', i.e., for solutions having large moduli, a linear (RHP) with variable coefficients. Global existence for q-connected dom ains was already obtained in [9] for the special case that the restric tion curves for the boundary data ''at infinity'' coincide with straig ht lines corresponding to linear (RHP)-s with special so-called consta nt-coefficient transversality boundary conditions. In this paper, the boundary conditions are much more general including highly nonlinear c onditions for bounded solutions in the context of nontransversality. I n order to prove global existence, we reduce the problem to nonlinear singular integral equations which can be treated by a degree theory of Fredholm-quasiruled mappings specifically constructed for mappings de fined by nonlinar pseudodifferential operators.