Title of article :
Gidas–Ni–Nirenberg results for finite difference
equations: Estimates of approximate symmetry
Author/Authors :
LD Humphreys and PJ McKenna ، نويسنده ,
Issue Information :
دوهفته نامه با شماره پیاپی سال 2007
Abstract :
Are positive solutions of finite difference boundary value problems Δhu = f (u) in Ωh, u = 0 on ∂Ωh
as symmetric as the domain? To answer this question we first show by examples that almost arbitrary nonsymmetric
solutions can be constructed. This is in striking difference to the continuous case, where by
the famous Gidas–Ni–Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related
problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243] positive solutions inherit
the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions
on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which
show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates
depends on whether or not f (0) < 0. The one-dimensional case stands out in two ways: the proofs are
elementary and the estimates for the defect of symmetry are O(h) compared to O(1/|log(h)|) in the higherdimensional
case.
© 2006 Elsevier Inc. All rights reserved.
Keywords :
Approximate symmetry , Finite difference equations , Nonlinear boundary value problems
Journal title :
Journal of Mathematical Analysis and Applications
Journal title :
Journal of Mathematical Analysis and Applications