2009/12/15 by Jonathan J. Sarhad, Sarhad, Jonathan J.
Engineering · Mathematics · #35J15 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #math.AP #msc:35J15
paper · pdf · doi:10.48550/arxiv.0912.2794
15 pages, 1 figure
arxiv created 2009/12/15 · openalex publication_date 2009/12/15 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article considers the semilinear boundary value problem given by the Poisson equation, -Δu=f(u) in a bounded domain Ω⊂ \Rn 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 Ω⊂ \R3, 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 H1(Ω) to the space of Hölder continuous functions Cα(Ω), 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 H2-regularity in the linear elliptic problem given by each iteration of Newton's method. We then use the regularity estimate to achieve convergence.