A Minimax Estimation by Incomplete Data in the Right-Hand Sides of Elliptic Second-Order Partial Differential Equations with Discontinuous Coefficients

Abstract

We consider systems described by boundary-value problems for elliptic second-order partial differential equations with discontinuous coefficients. These systems arise in the study of steady-state processes of a fluid filtration in multicomponent schistous mediums with nonuniform conditions for a nonideal contact. By observing the state of the systems, we find minimax estimates for functionals of the right-hand sides of these equations. Here we suppose that the right-hand sides of the equations, the boundary conditions, the junction conditions, and the error in the measurements are not known precisely, but we know only the sets to which they belong. We show that finding the minimax estimates is reduced to solving some integro-differential equations.