The constrained least gradient problem in 𝑅ⁿ

Source abstract

We consider the constrained least gradient problem $$inf { ∫ Ω | ∇ u | d x : u ∈ C 0 , 1 ( Ω ¯ ) , | ∇ u | ≤ 1 a.e. , u = g on ∂ Ω } \inf \left \{ {\int _\Omega {|\nabla u|dx:u \in {C^{0,1}}(\bar \Omega ),\quad |\nabla u| \leq 1\;{\text {a.e.}},u = g\;{\text {on}}\;\partial \Omega } } \right \}$$ 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.