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 non-symmetric 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 higher-dimensional case.

McKenna, P; Reichel, W (2007). *Gidas-Ni-Nirenberg results for finite difference equations: estimates of approximate symmetry.* Journal of Mathematical Analysis and Applications, 334(1):206-222.

## 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 non-symmetric 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 higher-dimensional case.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Approximate symmetry; Finite difference equations; Nonlinear boundary value problems |

Language: | English |

Date: | 2007 |

Deposited On: | 02 Nov 2009 12:19 |

Last Modified: | 05 Apr 2016 13:23 |

Publisher: | Elsevier |

ISSN: | 0022-247X |

Publisher DOI: | https://doi.org/10.1016/j.jmaa.2006.12.011 |

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