Coefficient Identification in Elliptic Partial Differential Equation

We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside of the unit square \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{D}$\end{document}, when over-posed boundary data are available. Following the main idea of the Method of Variational Imbedding (MVI), we “imbed” the inverse problem into a fourth-order elliptic boundary value problem for the Euler-Lagrange equation being the necessary condition for minimization of the quadratic functional of the original equation. The fourth-order boundary value problem becomes well-posed with the two boundary conditions considered here. The Euler-Lagrange equation for the unknown coefficient provides an explicit equation for the coefficient. A featuring example is elaborated numerically.