Quasilinear -equation on bordered Riemann surfaces

Abstract

We consider the existence of solutions of a nonlinear Riemann-Hilbert problem for a quasilinear ∂-equation on a bordered Riemann surface. (2000): Primary 35Q15, 30E25; Secondary 30G20 on such that u 0 (z) belongs to the bounded component of C\γ z for every z ∈ ∂ . Let K be a compact subset of and let ε > 0. Then there exists a smooth up to the boundary solution u of the equation (1.1) on such that u(z) ∈ γ z for every z ∈ ∂ and u − u 0 ∞ < ε on K. Mathematics Subject Classification

Some remarks on Hartogs’ extension lemma

“…Also, this is essentially the raison d'être for Bharali's class of functions in [2] to be that restrictive. An attempt to get such control via Riemann-Hilbert boundary value problems for quasilinear∂-equations on bordered Riemann surfaces yielded [3]. However the sup norm control one needs to conclude the proof of the generalization of Chirka's result over bordered Riemann surfaces with genus g > 0 seems to be still unknown.…”