Iterative bounds for the stable solutions of convex non-linear boundary value problems

Abstract

SynopsisWe consider a class of convex non-linear boundary value problems of the formwhere L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.

Constructive existence theorems for problems of Thomas‐Fermi Type

“…Following the notation and terminology used in [12] we study the non- where a E C1(D). a. E C(D) n K and K is the cone of non-negative real valued functions defined on 6.…”

“…y ( x ) ) = x-112 [y(x)I3l2 appearing in (1.1). Iterative procedures for general boundary value problems with convex nonlinearities have been obtained in Mooney and Roach [12] and are described in the following section.…”

“…with boundary conditions. coefficients and nonlinear function f appropriate to the application of existence results and iterative schemes contained in [12].…”

Exact bounds for the solution branches of nonlinear eigenvalue problems

“…In general, 2; consists of several branches some of which-in particular those made up by the minimal solutions _u(2)-are stable whereas the others are not. Stable branches are easy to obtain since the Picard iterates (and in the convex case also the Newton iterates, see Mooney-Roach [6]) converge monotonely. In the unstable case, however, the Picard sequence diverges and Newton's method requires good initial guesses (which are unknown, in general).…”

The dependence of critical parameter bounds on the monotonicity of a Newton sequence

“…This method uses the known fact that the Newton iterates {u,} starting with a suitable subsolution u 0 of (1.1), e.g. uo=0, converge monotonically upwards to the minimal positive solution u(2) whenever 2~A (Mooney/Roach [14]). Consequently a loss of monotonicity in the sequence {u,} implies that 2r…”