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Existence and uniqueness of minimizers of general least gradient problems.
- Source :
- Journal für die Reine und Angewandte Mathematik; Jan2018, Vol. 2017 Issue 734, p71-97, 27p
- Publication Year :
- 2018
-
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 inf<superscript>u∈BV<subscript>f</subscript>(Ω)</superscript>∫<subscript>Ω</subscript>φ(x,Du), where f:∂Ω→R is continuous, 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∈C<subscript>1,1</subscript>(Ω) is bounded away from zero, then minimizers of the weighted least gradient problem inf<subscript>u∈BV<subscript>f</subscript></subscript>∫<subscript>Ω</subscript>a∣Du∣ are unique in BVf(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∈C<superscript>1,α</superscript>(Ω) with any α<1.a∈C1,α(Ω) with any α<1. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00754102
- Volume :
- 2017
- Issue :
- 734
- Database :
- Complementary Index
- Journal :
- Journal für die Reine und Angewandte Mathematik
- Publication Type :
- Academic Journal
- Accession number :
- 127788770
- Full Text :
- https://doi.org/10.1515/ crelle-2014-0151