Abstract
We consider the classical problem of finding the density ϱ of a material body Ω embedded into a region S, when the potential generated by Ω (possibly coinciding with S) is known outside (or on the surface of) S. In the set of such solutions we look for the density\(\bar \varrho\)which has the smallestL2-norm and we prove that\(\bar \varrho\) belongs toL H 2= (Ω), the space of square summable functions harmonic in Ω. However\(\bar \varrho\)is unstable, i.e. itsL2-norm does not depend continuously upon the L2-norm of the potential. We show how a continuous dependence may be restored by introducing mild restrictions on the set of admissible solutions.
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