Existence, uniqueness, and regularity for functions of least gradient. (English) Zbl 0756.49021

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

J. Reine Angew. Math. 430, 35-60 (1992).

We investigate issues of existence, uniqueness and regularity for the least gradient problem \(\inf\{\|\nabla u\|(\Omega):\allowbreak u\in BV(\Omega)\cap C^ 0(\bar \Omega)\), \(u=g\text{ on }\partial\Omega\}\). Here \(\Omega\subset R^ n\) is a bounded Lipschitz domain, \(g: \partial \Omega\to R^ 1\) is continuous and \(\|\nabla u\|(\Omega)\) denotes the total variation of the vector-valued measure \(\nabla u\) evaluated on \(\Omega\). We give a constructive existence proof under a weak positive mean curvature hypothesis on \(\partial \Omega\), and show that without such an assumption there exist data \(g\) for which no solution exists. Under similar hypotheses, the solution is shown to be unique. Finally, we establish a modulus of continuity for the solution in terms of the modulus of continuity of the data. The techniques employed make extensive use of BV theory and sets of finite perimeter as well as certain maximum principles associated with area-minimizing hypersurfaces.

Reviewer: P.Sternberger

MSC:

49Q20

Variational problems in a geometric measure-theoretic setting

Keywords:

existence; uniqueness; regularity; least gradient problem; vector-valued measure; area-minimizing hypersurfaces