The inverse problems associated with these direct problems involve estimating the current sources within the volume conductor from measurements of voltages on the surface of either the head or body. Thus one would solve (3) with the boundary conditions:
The first is the Dirichlet condition, which says that one has a set of discrete measurements of the voltage of a subset of the outer surface. The second is the natural Neumann condition as seen before. While it doesn't look much different than the formulation of the direct problem, the mathematician Hadamard  noticed that inverse formulations of boundary value problems (or their algebraic counterpart) were often ill-behaved. He defined the conditions for well-posed and ill-posed problems.