Mathematics – Analysis of PDEs
Scientific paper
2009-12-15
Mathematics
Analysis of PDEs
15 pages, 1 figure
Scientific paper
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.
No associations
LandOfFree
The nonlinear Poisson equation via a Newton-imbedding procedure does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The nonlinear Poisson equation via a Newton-imbedding procedure, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The nonlinear Poisson equation via a Newton-imbedding procedure will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-304813