A Newton-imbedding procedure for solutions of semilinear boundary value problems in Sobolev spaces†

Abstract

This article is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of this method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H 1(Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity is also discussed. †Dedicated to Professor Guochun Wen on the occasion of his 70th birthday.