On some inverse problems for elliptic equations and systems

Abstract

Under study are the inverse problems of determining the right-hand side of a particular form and the solution for elliptic systems, including a series of elasticity systems. (On the boundary of the domain the solution satisfies either the Dirichlet conditions or mixed Dirichlet-Neumann conditions.) We assume that on a system of planes the normal derivatives of the solution can have discontinuities of the first kind. The conjugating boundary conditions on the discontinuity surface are analogous to the continuity conditions for the fields of displacements and stresses for a horizontally laminated medium. The overdetermination conditions are integral (the average of the solution over some domain is specified) or local (the values of the solution on some lines are specified). We study the solvability conditions for these problems and their Fredholm property.