Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems
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Abstract
We study the question of uniqueness of minimizers of the weighted least gradient problem where \(\int _{\Omega }|Dv|_a\) is the total variation with respect to the weight function a and S is the set of zeros of the function a. In contrast with previous results, which assume that the weight \(a\in C^{1,1}(\Omega )\) and is bounded away from zero, here a is only assumed to be continuous, and is allowed to vanish and also be discontinuous in certain subsets of \(\Omega \). We assume instead existence of a \(C^1\) minimizer. This problem arises naturally in the hybrid inverse problem of imaging electric conductivity from interior knowledge of the magnitude of one current density vector field, where existence of a \(C^1\) minimizer is known a priori.
$$\begin{aligned} \min \left\{ \int _{\Omega }|Dv|_a : v\in BV_{loc}(\Omega {\setminus } S),\; v|_{\partial \Omega }= f \right\} , \end{aligned}$$
Mathematics Subject Classification
35R30 35J60 31A25 62P10Notes
Acknowledgements
We would like to thank Robert L. Jerrard from whom we have learned a great deal throughout the course of this project. This work originated while the third author participated in the semester long Thematic Program on Inverse Problems and Imaging in the Fields Institute, January–May, 2012. The paper was essentially completed during the second authors participation in the program on Inverse Problems and Applications at the Mittag-Leffler Institute. We would like to thank both institutes for their hospitality and support. The first author was partially supported by MITACS and NSERC postdoctoral fellowships, and NSF Grant DMS-1715850. The second author was partially supported by an NSERC Discovery Grant. The third author was supported in part by the NSF Grant DMS-1312883. The authors would also like to thank the anonymous referee for careful reading of this paper and many useful comments.
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