The constrained least gradient problem in 𝑅ⁿ

Sternberg, Peter; Williams, Graham; Ziemer, William P.

doi:10.1090/S0002-9947-1993-1126213-2

The constrained least gradient problem in $\textbf {R}^ n$
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by Peter Sternberg, Graham Williams and William P. Ziemer

Trans. Amer. Math. Soc. 339 (1993), 403-432

DOI: https://doi.org/10.1090/S0002-9947-1993-1126213-2

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Abstract:

We consider the constrained least gradient problem \[ \inf \left \{ {\int _\Omega {|\nabla u|dx:u \in {C^{0,1}}(\bar \Omega ),\quad |\nabla u| \leq 1\;{\text {a.e.}},u = g\;{\text {on}}\;\partial \Omega } } \right \}\] 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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