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

The inverse problem of determining the unknown coefficient of the non-linear differential equation of torsional creep is studied. The unknown coefficient g = g(ξ2) depends on the gradient ξ := |∇u| of the solution u(x), x ∈ Ω ⊂ 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(Ω) of the solution u(x; g) of the direct problem with respect to the unknown coefficient g = g(ξ2) is obtained in the following sense: ‖u(x; g) − u(x; gm)‖1 → 0 when gm(η) → g(η) point-wise as m → ∞. 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.