Reconstructions from boundary measurements. (English) Zbl 0675.35084

Nachman, Adrian I.

Ann. Math. (2) 128, No. 3, 531-576 (1988).

The following problem is considered: Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\), \(n\geq 3\), with a \(C^{1,1}\) boundary and let \(\gamma\) (x) be a real-valued function in \(C^{1,1}({\bar \Omega})\) with a positive lower bound. Moreover, let \(u\in H^ 1(\Omega)\) be the unique solution of \[ \nabla \cdot (\gamma \nabla u)=0\quad in\quad \Omega,\quad u|_{\partial \Omega}=f, \] where \(f\in H^{1/2}(\partial \Omega)\) and let \(Q_{\gamma}(f)=\int_{\Omega}\gamma (x)| \nabla u(x)|^ 2 dx.\)
If \(\Omega\) represents an inhomogeneous, isotropic body with conductivity \(\gamma\), then \(Q_{\gamma}(f)\) is the power necessary to maintain a voltage potential f on the boundary \(\partial \Omega.\)
The author proofs that \(\gamma\) is uniquely determined by \(Q_{\gamma}\) and gives a solution to the reconstruction problem.

Reviewer: A.Neubauer

MSC:

35R30	Inverse problems for PDEs
35J25	Boundary value problems for second-order elliptic equations
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

identifiability; isotropic body; conductivity; voltage potential; uniquely determined; reconstruction problem