Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems

Abstract

We study the question of uniqueness of minimizers of the weighted least gradient problem $$\begin{aligned} \min \left\{ \int _{\Omega }|Dv|_a : v\in BV_{loc}(\Omega {\setminus } S),\; v|_{\partial \Omega }= f \right\} , \end{aligned}$$ 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.