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

Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems We study the question of uniqueness of minimizers of the weighted least gradient problem $$\begin{aligned} \min \left\{ \int _{\Omega }|Dv|_a : v\in BV_{loc}(\Omega {\setminus } S),\; v|_{\partial \Omega }= f \right\} , \end{aligned}$$ min ∫ Ω | D v | a : v ∈ B V loc ( Ω \ S ) , v | ∂ Ω = f , where $$\int _{\Omega }|Dv|_a$$ ∫ Ω | D v | 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 $$a\in C^{1,1}(\Omega )$$ a ∈ C 1 , 1 ( Ω ) 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 $$\Omega $$ Ω . We assume instead existence of a $$C^1$$ 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 $$C^1$$ C 1 minimizer is known a priori. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

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Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag GmbH Germany, part of Springer Nature
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
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