## Reconstructions from boundary measurements.(English)Zbl 0675.35084

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
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