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IMA Journal of Applied Mathematics Advance Access published online on November 17, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm056
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Determination of unknown coefficient in a non-linear elliptic problem related to the elastoplastic torsion of a bar

Alemdar Hasanov{dagger} and Arzu Erdem{ddagger}

Department of Mathematics, Kocaeli University, Umuttepe Kampusu, Izmit-Kocaeli 41380, Turkey

{dagger} Email: ahasanov{at}kou.edu.tr

{ddagger} Corresponding author. Email: aerdem{at}kou.edu.tr

Received on April 24, 2007; Accepted on October 10, 2007

The inverse problem of determining the unknown coefficient of the non-linear differential equation of torsional creep is studied. The unknown coefficient g = g({xi}2) depends on the gradient {xi}: = |{nabla}u| of the solution u(x), x isin {Omega} sub Rn, of the direct problem. It is proved that this gradient is bounded in C-norm. This permits one to choose the natural class of admissible coefficients for the considered inverse problem. The continuity in the norm of the Sobolev space H1({Omega}) of the solution u(x;g) of the direct problem with respect to the unknown coefficient g = g({xi}2) is obtained in the following sense: ||u(x;g) – u(x;gm)||1 -> 0 when gm({eta}) -> g({eta}) point-wise as m -> {infty}. Based on these results, the existence of a quasi-solution of the inverse problem in the considered class of admissible coefficients is obtained. Numerical examples related to determination of the unknown coefficient are presented.

Keywords: inverse coefficient problem; non-linear elliptic equation; torsional creep; existence of a quasi-solution.


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