This particular formulation is known as the Cauchy problem for Laplace's equation. At first glance it doesn't seem that we would gain much from this formulation; and in fact, it might appear that we are losing something in that we cannot easily infer the location of the sources that produced the voltages on the surface of the heart or skull if the sources are far from the surface. What we have gained is that the solution for such a problem is unique . The problem is still ill-posed in that the solution does not depend continuously on the data, but at least we do not have to contend with the multiplicity of solutions.
Let's explore this problem further. Note that we want to solve for the voltages on the inner (heart) surface given the voltages on the outer (thorax) surface, plus a Neuman boundary condition on the outer surface. If we represent this problem as a linear system (which we will do in detail in the section on Numerical Solution and Computational Considerations 4), we begin by expressing for the Cauchy problem in (10) as