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

Abstract

This paper 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 the 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 will also be discussed.AMS: 35J20, 35J60, 45B05, 65N30. KEY WORDS: nonlinear problem, Newton-imbedding iteration, variational formulation, boundary integral equations, a priori estimate.

The nonlinear Poisson equation via a Newton-embedding procedure

“…Remark: Since the C α norm has the L ∞ norm as a summand, Theorem 5.1 with p = ∞ suffices to show that a uniform bound on ||f (u)|| C α implies f is bounded. Therefore the assumptions made in [2], imply that f , f , and f are bounded functions. Moreover, under the same assumptions, as shown in the previous Section, f is linear.…”

“…has two parts. It is well described in [2], but recalled here for clarity. The procedure first imbeds the problem in a one-parameter family of problems,…”

“…and is again a model for (**). To facilitate the convergence estimates to follow, it will be helpful to use Taylor's theorem to simplify g. Similar to the application of a mean value theorem used in [2], for m > 1, g can be written as…”

“…−∆u = f (u) in Ω u| Γ = φ on Γ = ∂Ω, using the Newton-imbedding procedure that is applied in [2]. Here, f (u) is defined as f • u.…”

“…• 2. f and f are continuous maps from H 1 (Ω) to C α (Ω), α ∈ (0, 1 2 ]. An additional condition in [2] is the choice of a uniform width of time intervals in the procedure that ensures convergence, which exists as a consequence of the above assumptions. However, we prove the following theorems in Sections 2 and 3 of this article:…”