The nonlinear Poisson equation via a Newton-imbedding procedure

TL;DR

This paper analyzes the Newton-imbedding method for solving a nonlinear Poisson equation, proving conditions for solution existence, uniqueness, and regularity, and clarifying assumptions on the nonlinearity f.

Contribution

It demonstrates that certain boundedness assumptions on f' imply f must be constant, and establishes convergence and regularity results for the iterative solution process.

Findings

f must be constant under specific boundedness assumptions

Established convergence of the Newton-imbedding procedure

Proved H^2-regularity and uniqueness of solutions

Abstract

This article considers the semilinear boundary value problem given by the Poisson equation, -\Delta u=f(u) in a bounded domain \Omega\subset \R^{n} with smooth boundary. For the zero boundary value case, we approximate a solution using the Newton-imbedding procedure. With the assumptions that f, f', and f" are bounded functions on \R, with f'<0, and \Omega\subset \R^{3}, the Newton-imbedding procedure yields a continuous solution. This study is in response to an independent work which applies the same procedure, but assuming that f' maps the Sobolev space H^{1}(\Omega) to the space of H\"older continuous functions C^{\alpha}(\bar{\Omega}), and f(u), f'(u), and f"(u) have uniform bounds. In the first part of this article, we prove that these assumptions force f to be a constant function. In the remainder of the article, we prove the existence, uniqueness, and H^{2}-regularity in the linear elliptic problem given by each iteration of Newton's method. We then use the regularity estimate to achieve convergence.