An inverse coefficient problem for nonlinear biharmonic equation with surface measured data

A. Kh. Khasanov

Eurasian mathematical journal, Tome 1 (2010) no. 3, pp. 43-57

Résumé

An inverse coefficient problem for the nonlinear biharmonic equation $Au:=(g(\xi^2(u))(u_{x_1x_1}+(1/2)u_{x_2x_2})_{x_1x_1}+(g(\xi^2(u))u_{x_1x_2})_{x_1x_2}+(g(\xi^2(u))(u_{x_2x_2}+(1/2)u_{x_1x_1}))_{x_2x_2}=F(x)$, in $\Omega\subset R^2$, is considered. This problem arises in computational material science as a problem of identification of unknown properties of inelastic isotropic homogeneous incompressible bending plate using surface measured data. Within $J_2$-deformation theory of plasticity these properties are described by the coefficient $g(\xi^2(u))$ which depends on the effective value of the plate curvature: $\xi^2(u)=(u_{x_1x_1})^2+ (u_{x_2x_2})^2+ (u_{x_1x_2})^2+ u_{x_1x_1}u_{x_2x_2}$. The surface measured output data is assumed to be the deflections $w_i$, $i=\overline {1,M}$, at some points of the surface of a plate and obtained during the quasistatic process of bending. For a given coefficient $g(\xi^2(u))$ mathematical modeling of the bending problem leads to a nonlinear boundary value problem for the biharmonic equation with Dirichlet or mixed types of boundary conditions. Existence of the weak solution in the Sobolev space $H^2(\Omega)$ is proved by using the theory of monotone potential operators. A monotone iteration scheme for the linearized equation is proposed. Convergence in $H^2$-norm of the sequence of solutions of the linearized problem to the solution of the nonlinear problem is proved, and the rate of convergence is estimated. The obtained continuity property of the solution $u\in H^2(\Omega)$ of the direct problem, and compactness of the set of admissible coefficients $\mathcal {G}_0$ permit one to prove the existence of a quasi-solution of the considered inverse problem.