Combining The Runge Approximation And The Whitney Embedding Theorem In Hybrid Imaging
Résumé
This paper addresses enforcing non-vanishing constraints for solutions to a second order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension d ≥ 2, the family of 2d boundary conditions such that their Jacobian has maximal rank in the domain is both open and dense. Other constraints, which are relevant for applications to recent hybrid imaging modalities, are also considered. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [Greene and Wu, Ann. Inst. Fourier (Grenoble), 25(1, vii):215-235, 1975]. The method is very general, and can be used in other settings.
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