Inverse coefficient problems for monotone potential operators

Abstract

In this paper the class of inverse coefficient problems for nonlinear monotone potential elliptic operators is considered. This class is characterized by the property that the coefficient of elliptic operator depends on the gradient of the solution, i.e. on . The unknown coefficient is required to belong to a set of admissible coefficients which is compact in . Using a variational approach to the nonlinear direct problem it is shown that the solution of the linearized direct problem converges to the solution of the nonlinear direct problem in -norm. For the nonlinear direct problem weak -coefficient convergence is proved. This result allows one to prove the existence of quasisolutions of inverse problems with different types of additional conditions (measured data). As an important application of the theory, an inverse elastoplastic problem for a cylindrical bar and a nonlinear Sturm - Liouville problem are considered.