# A Tridimensional Inverse Shaping Problem

Abstract : We study a question which arises in the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Hamilton-Jacobi equation of eikonal type on the surface of the bubble whose solution is the norm of the magnetic induction field and which has a self-contained interest. We answer the question for closed smooth surfaces which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a ${rm C}^1$ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. But the condition mentioned above implies that not all analytic perturbed surfaces are shapable, as it can be seen by a counter-example.
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https://hal.inria.fr/inria-00074034
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Submitted on : Wednesday, May 24, 2006 - 2:21:38 PM
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### Identifiers

• HAL Id : inria-00074034, version 1

### Citation

Michel Pierre, Elisabeth Rouy. A Tridimensional Inverse Shaping Problem. [Research Report] RR-2655, INRIA. 1995, pp.34. ⟨inria-00074034⟩

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