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Existence and uniqueness of minimizers of general least gradient problems

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Published/Copyright: May 1, 2015
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 734

Abstract

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems – under certain sharp conditions – for minimizers of the general least gradient problem infu∈B⁢Vf⁢(Ω)⁡∫Ωφ⁢(x,D⁢u), where f:∂⁡Ω→ℝ is continuous, B⁢Vf⁢(Ω):={v∈B⁢V⁢(Ω):limr→0⁡ess⁢supy∈Ω,|x-y|<r⁡|f⁢(x)-v⁢(y)|=0⁢ for ⁢x∈∂⁡Ω} and φ⁢(x,ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular, we prove that if a∈C1,1⁢(Ω) is bounded away from zero, then minimizers of the weighted least gradient problem infu∈B⁢Vf⁡∫Ωa⁢|D⁢u| are unique in B⁢Vf⁢(Ω). We construct counterexamples to show that the regularity assumption a∈C1,1 is sharp, in the sense that it can not be replaced by a∈C1,α⁢(Ω) with any α<1.

Funding statement: The first and third author were partially supported by NSERC Discovery Grants. The second author was partially supported by an NSERC Postdoctoral Fellowship.

Acknowledgements

We are grateful to Ben Stephens for many helpful discussions. The third author wishes to thank the Mittag-Leffler Institute for their wonderful hospitality.

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Received: 2013-5-7
Published Online: 2015-5-1
Published in Print: 2018-1-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture
  3. Langlands program for p-adic coefficients and the petits camarades conjecture
  4. Existence and uniqueness of minimizers of general least gradient problems
  5. Partial regularity for mass-minimizing currents in Hilbert spaces
  6. Which abelian tensor categories are geometric?
  7. On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions
  8. Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow
  9. Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space
https://doi.org/10.1515/crelle-2014-0151

Articles in the same Issue

  1. Frontmatter
  2. The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture
  3. Langlands program for p-adic coefficients and the petits camarades conjecture
  4. Existence and uniqueness of minimizers of general least gradient problems
  5. Partial regularity for mass-minimizing currents in Hilbert spaces
  6. Which abelian tensor categories are geometric?
  7. On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions
  8. Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow
  9. Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space
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