On some inverse problems for elliptic equations and systems

We study the inverse problems for determining the right-hand side of a special form and the solution for elliptic systems, including a series of elasticity systems. On the boundary of the region the solution satisfies either the Dirichlet conditions or the mixed Dirichlet–Neuman conditions. On a set of planes we allow the normal derivatives of the solution to have discontinuities of the first kind. The gluing conditions on the discontinuity surface are analogous to the continuity conditions for the displacement and stress fields for horizontally layered medium. The overdetermination conditions are integral (the average of the solution over some region is given) or local (the solution is specified on some lines). For these problems we study solvability conditions and the Fredholm property.