A class of nonlinear boundary value problems for a second order elliptic system in general form

Abstract

We study a class of nonlinear boundary value problems BVP for second order elliptic systems. These nonlinear BVP were first introduced by L. V. Wolfersdorf in 5, 6 where a class of nonlinear BVP for holomorphic functions was studied and some efficient results derived. Later S. Knecht and A. S. Mashimba 2 and R. P. Gilbert and the authors of this paper [1, 4] made in depth investigations into a class of nonlinear BVP for first and second order elliptic systems. On the basis of these researches, this paper will further discuss a class of nonlinear BVP for second order systems in general form. First a transformation is introduced to convert the nonlinear BVP into a nonlinear generalized R H problem, and then, by use of corresponding integral operators and computation of the singular integrals, an equivalent nonlinear singular integral equation is presented. Finally, with techniques of singular integral operator theory, functional analysis, and function theoretic methods, under suitable hypotheses, solvability is proved for the boundary value problems