Recovery of the shape of interior objects from Cauchy data

We consider the boundary value problem
-∇(a.∇ u) + q u = F in Ω u = f on ∂Ω
in a bounded, smooth domain Ω⊂Rn containing a subdomain D⊂Ω with boundary ∂D in the class C². The functions a(x), q(x) and f(x) characterize the interior subdomain D and take on different constant values in D and in Ω/D.
a = a0 χ{D} + a1, q = q0 χ{D} + q1, F = F0 χ{D} + F1
where χ{D} is the characteristic function of the domain D. The values ao, a1, q0. . . need not be constant, but could be known functions of x - provided there is a discontinuity in the value at the boundary ∂D for at least one of the chracterising coefficients.

It is well known that for a given domain D, and assuming some smoothness on the function f, (say, f in H½(∂D), where Hs denote the usual Sobolev spaces of order s) there exists a unique solution u in H1/2(Ω) of this elliptic boundary value problem.

Here we are interested in the inverse problem consisting of the recovery of the shape of D from f plus the Neumann boundary values uν = g of the solution of the boundary value problem, where ν denotes the unit outward normal to ∂D.

There are several important sub-problems here. The most commonly attacked are the cases where there is only one coefficient characterising the domain D. Thus we might have a0=q0=0 but F0 > 0.

Depending on which coefficient is "active" there are a myriad of physical interpretation to this question. For example if the interior D is characterised by a material of different conductivity from the surrounding region Ω/D, then one would have a0 non zero. As a second example, we may want to recover the location and shape of an unknown heat source F. The strength of the source may only depend on the point x (in which case q0=0 but F0>0), or it might be proportional to the temperature u at x, that is we use the model F(x,t,u) = q(x) u(x,t) so that we have q0 non zero.

Our (this is joint work with Frank Hettlich) current interest lies with a time dependent version of this problem. Taking only the case of an interior source within an otherwise homogeneous region, the model we propose is to recover the shape and location of D from

ut - Γu = F := F0 χ{D} + F1 in Ω ×[0,T] u = f on ∂Ω×[0,T]
in a bounded cylindrical domain Ω×[0,T]. Overposed data consists of the flux or Neumann boundary values un(xj,t) = gj(t) for a finite number of boundary points xj, j=1, . . . N for all t in [0,T]. How many measurement points N are needed?

  • Frank Hettlich and William Rundell, Iterative Methods for the Reconstruction of an Inverse Potential Problem, Inverse Problems, 12, (1996), 251--266.

  • Frank Hettlich and William Rundell, Iterative Methods for the Recovery of the support of a source term in an elliptic differential equation, Inverse Problems, 13, (1997), 959-976.

  • Frank Hettlich and William Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14, (1998), 67-82.