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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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1274-x
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Dec 2, 2017

References

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