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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The constrained least gradient problem in $ {\bf R}\sp n$


Authors: Peter Sternberg, Graham Williams and William P. Ziemer
Journal: Trans. Amer. Math. Soc. 339 (1993), 403-432
MSC: Primary 49Q20
DOI: https://doi.org/10.1090/S0002-9947-1993-1126213-2
MathSciNet review: 1126213
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Abstract: We consider the constrained least gradient problem

$\displaystyle \inf \left\{ {\int_\Omega {\vert\nabla u\vert dx:u \in {C^{0,1}}(... ...a u\vert \leq 1\;{\text{a.e.}},u = g\;{\text{on}}\;\partial \Omega } } \right\}$

which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.

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DOI: https://doi.org/10.1090/S0002-9947-1993-1126213-2
Article copyright: © Copyright 1993 American Mathematical Society